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Theorem 2ndcctbss 17197
Description: If a topology is second-countable, every base has a countable subset which is a base. Exercise 16B2 in Willard. (Contributed by Jeff Hankins, 28-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Mar-2015.)
Hypotheses
Ref Expression
2ndcctbss.1  |-  X  = 
U. B
2ndcctbss.2  |-  J  =  ( topGen `  B )
2ndcctbss.3  |-  S  =  { <. u ,  v
>.  |  ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) ) }
Assertion
Ref Expression
2ndcctbss  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) )
Distinct variable groups:    b, c, u, v, w, B    J, b, c
Allowed substitution hints:    S( w, v, u, b, c)    J( w, v, u)    X( w, v, u, b, c)

Proof of Theorem 2ndcctbss
Dummy variables  d 
f  m  n  o  t  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . 3  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  J  e.  2ndc )
2 is2ndc 17188 . . 3  |-  ( J  e.  2ndc  <->  E. c  e.  TopBases  ( c  ~<_  om  /\  ( topGen `
 c )  =  J ) )
31, 2sylib 188 . 2  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  E. c  e. 
TopBases  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) )
4 vex 2804 . . . . . . . . 9  |-  c  e. 
_V
54, 4xpex 4817 . . . . . . . 8  |-  ( c  X.  c )  e. 
_V
6 3simpa 952 . . . . . . . . . 10  |-  ( ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) )  ->  ( u  e.  c  /\  v  e.  c ) )
76ssopab2i 4308 . . . . . . . . 9  |-  { <. u ,  v >.  |  ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) ) }  C_  { <. u ,  v >.  |  ( u  e.  c  /\  v  e.  c ) }
8 2ndcctbss.3 . . . . . . . . 9  |-  S  =  { <. u ,  v
>.  |  ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) ) }
9 df-xp 4711 . . . . . . . . 9  |-  ( c  X.  c )  =  { <. u ,  v
>.  |  ( u  e.  c  /\  v  e.  c ) }
107, 8, 93sstr4i 3230 . . . . . . . 8  |-  S  C_  ( c  X.  c
)
11 ssdomg 6923 . . . . . . . 8  |-  ( ( c  X.  c )  e.  _V  ->  ( S  C_  ( c  X.  c )  ->  S  ~<_  ( c  X.  c
) ) )
125, 10, 11mp2 17 . . . . . . 7  |-  S  ~<_  ( c  X.  c )
134xpdom1 6977 . . . . . . . . . . 11  |-  ( c  ~<_  om  ->  ( c  X.  c )  ~<_  ( om 
X.  c ) )
14 omex 7360 . . . . . . . . . . . 12  |-  om  e.  _V
1514xpdom2 6973 . . . . . . . . . . 11  |-  ( c  ~<_  om  ->  ( om  X.  c )  ~<_  ( om 
X.  om ) )
16 domtr 6930 . . . . . . . . . . 11  |-  ( ( ( c  X.  c
)  ~<_  ( om  X.  c )  /\  ( om  X.  c )  ~<_  ( om  X.  om )
)  ->  ( c  X.  c )  ~<_  ( om 
X.  om ) )
1713, 15, 16syl2anc 642 . . . . . . . . . 10  |-  ( c  ~<_  om  ->  ( c  X.  c )  ~<_  ( om 
X.  om ) )
18 xpomen 7659 . . . . . . . . . 10  |-  ( om 
X.  om )  ~~  om
19 domentr 6936 . . . . . . . . . 10  |-  ( ( ( c  X.  c
)  ~<_  ( om  X.  om )  /\  ( om  X.  om )  ~~  om )  ->  ( c  X.  c )  ~<_  om )
2017, 18, 19sylancl 643 . . . . . . . . 9  |-  ( c  ~<_  om  ->  ( c  X.  c )  ~<_  om )
2120adantr 451 . . . . . . . 8  |-  ( ( c  ~<_  om  /\  ( topGen `
 c )  =  J )  ->  (
c  X.  c )  ~<_  om )
2221ad2antll 709 . . . . . . 7  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( c  X.  c )  ~<_  om )
23 domtr 6930 . . . . . . 7  |-  ( ( S  ~<_  ( c  X.  c )  /\  (
c  X.  c )  ~<_  om )  ->  S  ~<_  om )
2412, 22, 23sylancr 644 . . . . . 6  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  S  ~<_  om )
258relopabi 4827 . . . . . . . . . . 11  |-  Rel  S
26 simpr 447 . . . . . . . . . . 11  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  x  e.  S )
27 1st2nd 6182 . . . . . . . . . . 11  |-  ( ( Rel  S  /\  x  e.  S )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2825, 26, 27sylancr 644 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  x  =  <. ( 1st `  x
) ,  ( 2nd `  x ) >. )
2928, 26eqeltrrd 2371 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  S )
30 df-br 4040 . . . . . . . . . . 11  |-  ( ( 1st `  x ) S ( 2nd `  x
)  <->  <. ( 1st `  x
) ,  ( 2nd `  x ) >.  e.  S
)
31 fvex 5555 . . . . . . . . . . . 12  |-  ( 1st `  x )  e.  _V
32 fvex 5555 . . . . . . . . . . . 12  |-  ( 2nd `  x )  e.  _V
33 simpl 443 . . . . . . . . . . . . . 14  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  u  =  ( 1st `  x
) )
3433eleq1d 2362 . . . . . . . . . . . . 13  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
u  e.  c  <->  ( 1st `  x )  e.  c ) )
35 simpr 447 . . . . . . . . . . . . . 14  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  v  =  ( 2nd `  x
) )
3635eleq1d 2362 . . . . . . . . . . . . 13  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
v  e.  c  <->  ( 2nd `  x )  e.  c ) )
37 sseq1 3212 . . . . . . . . . . . . . . 15  |-  ( u  =  ( 1st `  x
)  ->  ( u  C_  w  <->  ( 1st `  x
)  C_  w )
)
38 sseq2 3213 . . . . . . . . . . . . . . 15  |-  ( v  =  ( 2nd `  x
)  ->  ( w  C_  v  <->  w  C_  ( 2nd `  x ) ) )
3937, 38bi2anan9 843 . . . . . . . . . . . . . 14  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
( u  C_  w  /\  w  C_  v )  <-> 
( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) ) )
4039rexbidv 2577 . . . . . . . . . . . . 13  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  ( E. w  e.  B  ( u  C_  w  /\  w  C_  v )  <->  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) ) )
4134, 36, 403anbi123d 1252 . . . . . . . . . . . 12  |-  ( ( u  =  ( 1st `  x )  /\  v  =  ( 2nd `  x
) )  ->  (
( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v
) )  <->  ( ( 1st `  x )  e.  c  /\  ( 2nd `  x )  e.  c  /\  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) ) ) )
4231, 32, 41, 8braba 4298 . . . . . . . . . . 11  |-  ( ( 1st `  x ) S ( 2nd `  x
)  <->  ( ( 1st `  x )  e.  c  /\  ( 2nd `  x
)  e.  c  /\  E. w  e.  B  ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) ) )
4330, 42bitr3i 242 . . . . . . . . . 10  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  S  <->  ( ( 1st `  x
)  e.  c  /\  ( 2nd `  x )  e.  c  /\  E. w  e.  B  (
( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) ) )
4443simp3bi 972 . . . . . . . . 9  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  S  ->  E. w  e.  B  ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) )
4529, 44syl 15 . . . . . . . 8  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) )
46 fvi 5595 . . . . . . . . . 10  |-  ( B  e.  TopBases  ->  (  _I  `  B )  =  B )
4746ad3antrrr 710 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  (  _I  `  B )  =  B )
4847rexeqdv 2756 . . . . . . . 8  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  ( E. w  e.  (  _I  `  B ) ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) )  <->  E. w  e.  B  ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) ) )
4945, 48mpbird 223 . . . . . . 7  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  x  e.  S )  ->  E. w  e.  (  _I  `  B
) ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) )
5049ralrimiva 2639 . . . . . 6  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  A. x  e.  S  E. w  e.  (  _I  `  B ) ( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) ) )
51 fvex 5555 . . . . . . 7  |-  (  _I 
`  B )  e. 
_V
52 sseq2 3213 . . . . . . . 8  |-  ( w  =  ( f `  x )  ->  (
( 1st `  x
)  C_  w  <->  ( 1st `  x )  C_  (
f `  x )
) )
53 sseq1 3212 . . . . . . . 8  |-  ( w  =  ( f `  x )  ->  (
w  C_  ( 2nd `  x )  <->  ( f `  x )  C_  ( 2nd `  x ) ) )
5452, 53anbi12d 691 . . . . . . 7  |-  ( w  =  ( f `  x )  ->  (
( ( 1st `  x
)  C_  w  /\  w  C_  ( 2nd `  x
) )  <->  ( ( 1st `  x )  C_  ( f `  x
)  /\  ( f `  x )  C_  ( 2nd `  x ) ) ) )
5551, 54axcc4dom 8083 . . . . . 6  |-  ( ( S  ~<_  om  /\  A. x  e.  S  E. w  e.  (  _I  `  B
) ( ( 1st `  x )  C_  w  /\  w  C_  ( 2nd `  x ) ) )  ->  E. f ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )
5624, 50, 55syl2anc 642 . . . . 5  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  E. f ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )
5746ad2antrr 706 . . . . . . . . 9  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  (  _I  `  B )  =  B )
58 feq3 5393 . . . . . . . . 9  |-  ( (  _I  `  B )  =  B  ->  (
f : S --> (  _I 
`  B )  <->  f : S
--> B ) )
5957, 58syl 15 . . . . . . . 8  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( f : S --> (  _I  `  B )  <->  f : S
--> B ) )
6059anbi1d 685 . . . . . . 7  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  <->  ( f : S --> B  /\  A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) ) )
61 2ndctop 17189 . . . . . . . . . . . . . 14  |-  ( J  e.  2ndc  ->  J  e. 
Top )
6261adantl 452 . . . . . . . . . . . . 13  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  J  e.  Top )
6362ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  J  e.  Top )
64 frn 5411 . . . . . . . . . . . . . 14  |-  ( f : S --> B  ->  ran  f  C_  B )
6564ad2antrl 708 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  C_  B )
66 bastg 16720 . . . . . . . . . . . . . . 15  |-  ( B  e.  TopBases  ->  B  C_  ( topGen `
 B ) )
6766ad3antrrr 710 . . . . . . . . . . . . . 14  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  B  C_  ( topGen `
 B ) )
68 2ndcctbss.2 . . . . . . . . . . . . . 14  |-  J  =  ( topGen `  B )
6967, 68syl6sseqr 3238 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  B  C_  J
)
7065, 69sstrd 3202 . . . . . . . . . . . 12  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  C_  J )
71 simprrl 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  o  e.  J
)
72 simprr 733 . . . . . . . . . . . . . . . . . 18  |-  ( ( c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) )  -> 
( topGen `  c )  =  J )
7372ad2antlr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  ( topGen `  c
)  =  J )
7471, 73eleqtrrd 2373 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  o  e.  (
topGen `  c ) )
75 simprrr 741 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  t  e.  o )
76 tg2 16719 . . . . . . . . . . . . . . . 16  |-  ( ( o  e.  ( topGen `  c )  /\  t  e.  o )  ->  E. d  e.  c  ( t  e.  d  /\  d  C_  o ) )
7774, 75, 76syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  E. d  e.  c  ( t  e.  d  /\  d  C_  o
) )
78 bastg 16720 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( c  e.  TopBases  ->  c  C_  ( topGen `
 c ) )
7978ad2antrl 708 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  c  C_  ( topGen `
 c ) )
8079ad2antrr 706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  c  C_  ( topGen `
 c ) )
8168eqeq2i 2306 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
topGen `  c )  =  J  <->  ( topGen `  c
)  =  ( topGen `  B ) )
8281biimpi 186 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
topGen `  c )  =  J  ->  ( topGen `  c )  =  (
topGen `  B ) )
8382adantl 452 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( c  ~<_  om  /\  ( topGen `
 c )  =  J )  ->  ( topGen `
 c )  =  ( topGen `  B )
)
8483ad2antll 709 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( topGen `  c
)  =  ( topGen `  B ) )
8584ad2antrr 706 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  ( topGen `  c
)  =  ( topGen `  B ) )
8680, 85sseqtrd 3227 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  c  C_  ( topGen `
 B ) )
87 simprl 732 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  d  e.  c )
8886, 87sseldd 3194 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  d  e.  (
topGen `  B ) )
89 simprrl 740 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  t  e.  d )
90 tg2 16719 . . . . . . . . . . . . . . . . . . 19  |-  ( ( d  e.  ( topGen `  B )  /\  t  e.  d )  ->  E. m  e.  B  ( t  e.  m  /\  m  C_  d ) )
9188, 89, 90syl2anc 642 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  E. m  e.  B  ( t  e.  m  /\  m  C_  d ) )
9266ad3antrrr 710 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  B  C_  ( topGen `
 B ) )
9392ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  B  C_  ( topGen `
 B ) )
9473ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  ( topGen `  c
)  =  J )
9594, 68syl6req 2345 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  ( topGen `  B
)  =  ( topGen `  c ) )
9693, 95sseqtrd 3227 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  B  C_  ( topGen `
 c ) )
97 simprl 732 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  m  e.  B
)
9896, 97sseldd 3194 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  m  e.  (
topGen `  c ) )
99 simprrl 740 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  t  e.  m
)
100 tg2 16719 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( m  e.  ( topGen `  c )  /\  t  e.  m )  ->  E. n  e.  c  ( t  e.  n  /\  n  C_  m ) )
10198, 99, 100syl2anc 642 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  E. n  e.  c  ( t  e.  n  /\  n  C_  m ) )
102 ffn 5405 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( f : S --> B  -> 
f  Fn  S )
103102ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) )  -> 
f  Fn  S )
104103ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  f  Fn  S
)
105104ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  f  Fn  S
)
106 simprl 732 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  e.  c )
10787ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  d  e.  c )
108 simplrl 736 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  e.  B
)
109 simprrr 741 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  C_  m
)
110 simprr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) )  ->  m  C_  d
)
111110ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  C_  d
)
112 sseq2 3213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( w  =  m  ->  (
n  C_  w  <->  n  C_  m
) )
113 sseq1 3212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( w  =  m  ->  (
w  C_  d  <->  m  C_  d
) )
114112, 113anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( w  =  m  ->  (
( n  C_  w  /\  w  C_  d )  <-> 
( n  C_  m  /\  m  C_  d ) ) )
115114rspcev 2897 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( m  e.  B  /\  ( n  C_  m  /\  m  C_  d ) )  ->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) )
116108, 109, 111, 115syl12anc 1180 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) )
117 df-br 4040 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( n S d  <->  <. n ,  d >.  e.  S
)
118 vex 2804 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  n  e. 
_V
119 vex 2804 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  d  e. 
_V
120 simpl 443 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( u  =  n  /\  v  =  d )  ->  u  =  n )
121120eleq1d 2362 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( u  =  n  /\  v  =  d )  ->  ( u  e.  c  <-> 
n  e.  c ) )
122 simpr 447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( u  =  n  /\  v  =  d )  ->  v  =  d )
123122eleq1d 2362 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( u  =  n  /\  v  =  d )  ->  ( v  e.  c  <-> 
d  e.  c ) )
124 sseq1 3212 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( u  =  n  ->  (
u  C_  w  <->  n  C_  w
) )
125 sseq2 3213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( v  =  d  ->  (
w  C_  v  <->  w  C_  d
) )
126124, 125bi2anan9 843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( u  =  n  /\  v  =  d )  ->  ( ( u  C_  w  /\  w  C_  v
)  <->  ( n  C_  w  /\  w  C_  d
) ) )
127126rexbidv 2577 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( u  =  n  /\  v  =  d )  ->  ( E. w  e.  B  ( u  C_  w  /\  w  C_  v
)  <->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) )
128121, 123, 1273anbi123d 1252 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( u  =  n  /\  v  =  d )  ->  ( ( u  e.  c  /\  v  e.  c  /\  E. w  e.  B  ( u  C_  w  /\  w  C_  v ) )  <->  ( n  e.  c  /\  d  e.  c  /\  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) ) )
129118, 119, 128, 8braba 4298 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( n S d  <->  ( n  e.  c  /\  d  e.  c  /\  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) )
130117, 129bitr3i 242 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( <.
n ,  d >.  e.  S  <->  ( n  e.  c  /\  d  e.  c  /\  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) ) )
131106, 107, 116, 130syl3anbrc 1136 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  <. n ,  d
>.  e.  S )
132 fnfvelrn 5678 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( f  Fn  S  /\  <.
n ,  d >.  e.  S )  ->  (
f `  <. n ,  d >. )  e.  ran  f )
133105, 131, 132syl2anc 642 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( f `  <. n ,  d >.
)  e.  ran  f
)
134 simprl 732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  e.  c )
135 simplll 734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  d  e.  c )
136 simplrl 736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  e.  B )
137 simprrr 741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  n  C_  m )
138110ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  m  C_  d )
139136, 137, 138, 115syl12anc 1180 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  E. w  e.  B  ( n  C_  w  /\  w  C_  d ) )
140134, 135, 139, 130syl3anbrc 1136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  <. n ,  d >.  e.  S
)
141 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( x  =  <. n ,  d
>.  ->  ( 1st `  x
)  =  ( 1st `  <. n ,  d
>. ) )
142 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( x  =  <. n ,  d
>.  ->  ( f `  x )  =  ( f `  <. n ,  d >. )
)
143141, 142sseq12d 3220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( x  =  <. n ,  d
>.  ->  ( ( 1st `  x )  C_  (
f `  x )  <->  ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )
) )
144 fveq2 5541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( x  =  <. n ,  d
>.  ->  ( 2nd `  x
)  =  ( 2nd `  <. n ,  d
>. ) )
145142, 144sseq12d 3220 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( x  =  <. n ,  d
>.  ->  ( ( f `
 x )  C_  ( 2nd `  x )  <-> 
( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
) )
146143, 145anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( x  =  <. n ,  d
>.  ->  ( ( ( 1st `  x ) 
C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  <->  ( ( 1st `  <. n ,  d
>. )  C_  ( f `
 <. n ,  d
>. )  /\  (
f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d
>. ) ) ) )
147146rspcv 2893 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( <.
n ,  d >.  e.  S  ->  ( A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
) ) )
148140, 147syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
) ) )
149118, 119op1st 6144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( 1st `  <. n ,  d
>. )  =  n
150149sseq1i 3215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  <->  n 
C_  ( f `  <. n ,  d >.
) )
151118, 119op2nd 6145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( 2nd `  <. n ,  d
>. )  =  d
152151sseq2i 3216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )  <->  ( f `  <. n ,  d >. )  C_  d )
153150, 152anbi12i 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
)  <->  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )
154 simprl 732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  ->  n  C_  ( f `  <. n ,  d >.
) )
155 simprl 732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) )  ->  t  e.  n )
156155ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
t  e.  n )
157154, 156sseldd 3194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
t  e.  ( f `
 <. n ,  d
>. ) )
158 simprr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
( f `  <. n ,  d >. )  C_  d )
159 simplrr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35  |-  ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  d  C_  o )
160159ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
d  C_  o )
161158, 160sstrd 3202 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
( f `  <. n ,  d >. )  C_  o )
162157, 161jca 518 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32  |-  ( ( ( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  /\  ( n  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  d ) )  -> 
( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) )
163162ex 423 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  (
( n  C_  (
f `  <. n ,  d >. )  /\  (
f `  <. n ,  d >. )  C_  d
)  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
164153, 163syl5bi 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  (
( ( 1st `  <. n ,  d >. )  C_  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  ( 2nd `  <. n ,  d >. )
)  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
165148, 164syld 40 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  (
n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
t  e.  ( f `
 <. n ,  d
>. )  /\  (
f `  <. n ,  d >. )  C_  o
) ) )
166165com12 27 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
167166exp4c 591 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( A. x  e.  S  (
( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) )  ->  (
( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  -> 
( ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) )  -> 
( ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) )  -> 
( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) ) ) )
168167ad2antlr 707 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) )  -> 
( ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) )  -> 
( ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) )  -> 
( ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) )  -> 
( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) ) ) )
169168adantl 452 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  ( ( d  e.  c  /\  (
t  e.  d  /\  d  C_  o ) )  ->  ( ( m  e.  B  /\  (
t  e.  m  /\  m  C_  d ) )  ->  ( ( n  e.  c  /\  (
t  e.  n  /\  n  C_  m ) )  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) ) ) )
170169imp41 576 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) )
171 eleq2 2357 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( b  =  ( f `  <. n ,  d >.
)  ->  ( t  e.  b  <->  t  e.  ( f `  <. n ,  d >. )
) )
172 sseq1 3212 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( b  =  ( f `  <. n ,  d >.
)  ->  ( b  C_  o  <->  ( f `  <. n ,  d >.
)  C_  o )
)
173171, 172anbi12d 691 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( b  =  ( f `  <. n ,  d >.
)  ->  ( (
t  e.  b  /\  b  C_  o )  <->  ( t  e.  ( f `  <. n ,  d >. )  /\  ( f `  <. n ,  d >. )  C_  o ) ) )
174173rspcev 2897 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( f `  <. n ,  d >. )  e.  ran  f  /\  (
t  e.  ( f `
 <. n ,  d
>. )  /\  (
f `  <. n ,  d >. )  C_  o
) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
175133, 170, 174syl2anc 642 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  /\  ( n  e.  c  /\  ( t  e.  n  /\  n  C_  m ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
176175exp32 588 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  ( n  e.  c  ->  ( (
t  e.  n  /\  n  C_  m )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) ) )
177176rexlimdv 2679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  ( E. n  e.  c  ( t  e.  n  /\  n  C_  m )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) )
178101, 177mpd 14 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  (
c  e.  TopBases  /\  (
c  ~<_  om  /\  ( topGen `
 c )  =  J ) ) )  /\  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  /\  ( o  e.  J  /\  t  e.  o
) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  /\  ( m  e.  B  /\  ( t  e.  m  /\  m  C_  d ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
179178exp32 588 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  ( m  e.  B  ->  ( (
t  e.  m  /\  m  C_  d )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) ) )
180179rexlimdv 2679 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  ( E. m  e.  B  ( t  e.  m  /\  m  C_  d )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) )
18191, 180mpd 14 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  /\  ( d  e.  c  /\  ( t  e.  d  /\  d  C_  o ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
182181exp32 588 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  ( d  e.  c  ->  ( (
t  e.  d  /\  d  C_  o )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) ) )
183182rexlimdv 2679 . . . . . . . . . . . . . . 15  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  ( E. d  e.  c  ( t  e.  d  /\  d  C_  o )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) )
18477, 183mpd 14 . . . . . . . . . . . . . 14  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  /\  ( o  e.  J  /\  t  e.  o ) ) )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
185184expr 598 . . . . . . . . . . . . 13  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ( ( o  e.  J  /\  t  e.  o )  ->  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) ) )
186185ralrimivv 2647 . . . . . . . . . . . 12  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  A. o  e.  J  A. t  e.  o  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )
187 basgen2 16743 . . . . . . . . . . . 12  |-  ( ( J  e.  Top  /\  ran  f  C_  J  /\  A. o  e.  J  A. t  e.  o  E. b  e.  ran  f ( t  e.  b  /\  b  C_  o ) )  ->  ( topGen `  ran  f )  =  J )
18863, 70, 186, 187syl3anc 1182 . . . . . . . . . . 11  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ( topGen `  ran  f )  =  J )
189188, 63eqeltrd 2370 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ( topGen `  ran  f )  e.  Top )
190 tgclb 16724 . . . . . . . . . 10  |-  ( ran  f  e.  TopBases  <->  ( topGen ` 
ran  f )  e. 
Top )
191189, 190sylibr 203 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  e.  TopBases )
192 omelon 7363 . . . . . . . . . . . 12  |-  om  e.  On
19324adantr 451 . . . . . . . . . . . 12  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  S  ~<_  om )
194 ondomen 7680 . . . . . . . . . . . 12  |-  ( ( om  e.  On  /\  S  ~<_  om )  ->  S  e.  dom  card )
195192, 193, 194sylancr 644 . . . . . . . . . . 11  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  S  e.  dom  card )
196102ad2antrl 708 . . . . . . . . . . . 12  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  f  Fn  S
)
197 dffn4 5473 . . . . . . . . . . . 12  |-  ( f  Fn  S  <->  f : S -onto-> ran  f )
198196, 197sylib 188 . . . . . . . . . . 11  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  f : S -onto-> ran  f )
199 fodomnum 7700 . . . . . . . . . . 11  |-  ( S  e.  dom  card  ->  ( f : S -onto-> ran  f  ->  ran  f  ~<_  S ) )
200195, 198, 199sylc 56 . . . . . . . . . 10  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  ~<_  S )
201 domtr 6930 . . . . . . . . . 10  |-  ( ( ran  f  ~<_  S  /\  S  ~<_  om )  ->  ran  f  ~<_  om )
202200, 193, 201syl2anc 642 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  ran  f  ~<_  om )
203188eqcomd 2301 . . . . . . . . 9  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  J  =  (
topGen `  ran  f ) )
204 breq1 4042 . . . . . . . . . . 11  |-  ( b  =  ran  f  -> 
( b  ~<_  om  <->  ran  f  ~<_  om ) )
205 sseq1 3212 . . . . . . . . . . 11  |-  ( b  =  ran  f  -> 
( b  C_  B  <->  ran  f  C_  B )
)
206 fveq2 5541 . . . . . . . . . . . 12  |-  ( b  =  ran  f  -> 
( topGen `  b )  =  ( topGen `  ran  f ) )
207206eqeq2d 2307 . . . . . . . . . . 11  |-  ( b  =  ran  f  -> 
( J  =  (
topGen `  b )  <->  J  =  ( topGen `  ran  f ) ) )
208204, 205, 2073anbi123d 1252 . . . . . . . . . 10  |-  ( b  =  ran  f  -> 
( ( b  ~<_  om 
/\  b  C_  B  /\  J  =  ( topGen `
 b ) )  <-> 
( ran  f  ~<_  om  /\  ran  f  C_  B  /\  J  =  ( topGen ` 
ran  f ) ) ) )
209208rspcev 2897 . . . . . . . . 9  |-  ( ( ran  f  e.  TopBases  /\  ( ran  f  ~<_  om  /\  ran  f  C_  B  /\  J  =  ( topGen ` 
ran  f ) ) )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) )
210191, 202, 65, 203, 209syl13anc 1184 . . . . . . . 8  |-  ( ( ( ( B  e.  TopBases 
/\  J  e.  2ndc )  /\  ( c  e.  TopBases 
/\  ( c  ~<_  om 
/\  ( topGen `  c
)  =  J ) ) )  /\  (
f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b )
) )
211210ex 423 . . . . . . 7  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( ( f : S --> B  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  (
topGen `  b ) ) ) )
21260, 211sylbid 206 . . . . . 6  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( ( f : S --> (  _I 
`  B )  /\  A. x  e.  S  ( ( 1st `  x
)  C_  ( f `  x )  /\  (
f `  x )  C_  ( 2nd `  x
) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  (
topGen `  b ) ) ) )
213212exlimdv 1626 . . . . 5  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  ( E. f
( f : S --> (  _I  `  B )  /\  A. x  e.  S  ( ( 1st `  x )  C_  (
f `  x )  /\  ( f `  x
)  C_  ( 2nd `  x ) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b )
) ) )
21456, 213mpd 14 . . . 4  |-  ( ( ( B  e.  TopBases  /\  J  e.  2ndc )  /\  ( c  e.  TopBases  /\  ( c  ~<_  om  /\  ( topGen `  c )  =  J ) ) )  ->  E. b  e.  TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b )
) )
215214exp32 588 . . 3  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  (
c  e.  TopBases  ->  (
( c  ~<_  om  /\  ( topGen `  c )  =  J )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) ) ) )
216215rexlimdv 2679 . 2  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  ( E. c  e.  TopBases  ( c  ~<_  om  /\  ( topGen `  c )  =  J )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) ) )
2173, 216mpd 14 1  |-  ( ( B  e.  TopBases  /\  J  e.  2ndc )  ->  E. b  e. 
TopBases  ( b  ~<_  om  /\  b  C_  B  /\  J  =  ( topGen `  b
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   <.cop 3656   U.cuni 3843   class class class wbr 4039   {copab 4092    _I cid 4320   Oncon0 4408   omcom 4672    X. cxp 4703   dom cdm 4705   ran crn 4706   Rel wrel 4710    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271   1stc1st 6136   2ndc2nd 6137    ~~ cen 6876    ~<_ cdom 6877   cardccrd 7584   topGenctg 13358   Topctop 16647   TopBasesctb 16651   2ndcc2ndc 17180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cc 8077
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-card 7588  df-acn 7591  df-topgen 13360  df-top 16652  df-bases 16654  df-2ndc 17182
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