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Theorem pwfseqlem4 8463
Description: Lemma for pwfseq 8465. Derive a final contradiction from the function  F in pwfseqlem3 8461. Applying fpwwe2 8444 to it, we get a certain maximal well-ordered subset 
Z, but the defining property  ( Z F ( W `  Z
) )  e.  Z contradicts our assumption on  F, so we are reduced to the case of 
Z finite. This too is a contradiction, though, because  Z and its preimage under  ( W `  Z
) are distinct sets of the same cardinality and in a subset relation, which is impossible for finite sets. (Contributed by Mario Carneiro, 31-May-2015.)
Hypotheses
Ref Expression
pwfseqlem4.g  |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )
pwfseqlem4.x  |-  ( ph  ->  X  C_  A )
pwfseqlem4.h  |-  ( ph  ->  H : om -1-1-onto-> X )
pwfseqlem4.ps  |-  ( ps  <->  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )
pwfseqlem4.k  |-  ( (
ph  /\  ps )  ->  K : U_ n  e.  om  ( x  ^m  n ) -1-1-> x )
pwfseqlem4.d  |-  D  =  ( G `  {
w  e.  x  |  ( ( `' K `  w )  e.  ran  G  /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )
pwfseqlem4.f  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
pwfseqlem4.w  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. b  e.  a 
[. ( `' s
" { b } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  b ) ) }
pwfseqlem4.z  |-  Z  = 
U. dom  W
Assertion
Ref Expression
pwfseqlem4  |-  -.  ph
Distinct variable groups:    n, r, w, x, z    D, n, z    a, b, s, v, F    w, G    w, K    r, a, x, z, H, b, s, v    n, a, ph, b, s, v, r, x, z    ps, n, z    A, a, n, r, s, x, z    W, a, b, s, v    Z, a, b, s, v
Allowed substitution hints:    ph( w)    ps( x, w, v, s, r, a, b)    A( w, v, b)    D( x, w, v, s, r, a, b)    F( x, z, w, n, r)    G( x, z, v, n, s, r, a, b)    H( w, n)    K( x, z, v, n, s, r, a, b)    W( x, z, w, n, r)    X( x, z, w, v, n, s, r, a, b)    Z( x, z, w, n, r)

Proof of Theorem pwfseqlem4
StepHypRef Expression
1 eqid 2380 . . . . . . . . . . 11  |-  Z  =  Z
2 eqid 2380 . . . . . . . . . . 11  |-  ( W `
 Z )  =  ( W `  Z
)
31, 2pm3.2i 442 . . . . . . . . . 10  |-  ( Z  =  Z  /\  ( W `  Z )  =  ( W `  Z ) )
4 pwfseqlem4.w . . . . . . . . . . 11  |-  W  =  { <. a ,  s
>.  |  ( (
a  C_  A  /\  s  C_  ( a  X.  a ) )  /\  ( s  We  a  /\  A. b  e.  a 
[. ( `' s
" { b } )  /  v ]. ( v F ( s  i^i  ( v  X.  v ) ) )  =  b ) ) }
5 pwfseqlem4.g . . . . . . . . . . . . 13  |-  ( ph  ->  G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n ) )
6 omex 7524 . . . . . . . . . . . . . 14  |-  om  e.  _V
7 ovex 6038 . . . . . . . . . . . . . 14  |-  ( A  ^m  n )  e. 
_V
86, 7iunex 5923 . . . . . . . . . . . . 13  |-  U_ n  e.  om  ( A  ^m  n )  e.  _V
9 f1dmex 5903 . . . . . . . . . . . . 13  |-  ( ( G : ~P A -1-1-> U_ n  e.  om  ( A  ^m  n )  /\  U_ n  e.  om  ( A  ^m  n )  e. 
_V )  ->  ~P A  e.  _V )
105, 8, 9sylancl 644 . . . . . . . . . . . 12  |-  ( ph  ->  ~P A  e.  _V )
11 pwexb 4686 . . . . . . . . . . . 12  |-  ( A  e.  _V  <->  ~P A  e.  _V )
1210, 11sylibr 204 . . . . . . . . . . 11  |-  ( ph  ->  A  e.  _V )
13 pwfseqlem4.x . . . . . . . . . . . 12  |-  ( ph  ->  X  C_  A )
14 pwfseqlem4.h . . . . . . . . . . . 12  |-  ( ph  ->  H : om -1-1-onto-> X )
15 pwfseqlem4.ps . . . . . . . . . . . 12  |-  ( ps  <->  ( ( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  /\  om  ~<_  x ) )
16 pwfseqlem4.k . . . . . . . . . . . 12  |-  ( (
ph  /\  ps )  ->  K : U_ n  e.  om  ( x  ^m  n ) -1-1-> x )
17 pwfseqlem4.d . . . . . . . . . . . 12  |-  D  =  ( G `  {
w  e.  x  |  ( ( `' K `  w )  e.  ran  G  /\  -.  w  e.  ( `' G `  ( `' K `  w ) ) ) } )
18 pwfseqlem4.f . . . . . . . . . . . 12  |-  F  =  ( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
195, 13, 14, 15, 16, 17, 18pwfseqlem4a 8462 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( a F s )  e.  A )
20 pwfseqlem4.z . . . . . . . . . . 11  |-  Z  = 
U. dom  W
214, 12, 19, 20fpwwe2 8444 . . . . . . . . . 10  |-  ( ph  ->  ( ( Z W ( W `  Z
)  /\  ( Z F ( W `  Z ) )  e.  Z )  <->  ( Z  =  Z  /\  ( W `  Z )  =  ( W `  Z ) ) ) )
223, 21mpbiri 225 . . . . . . . . 9  |-  ( ph  ->  ( Z W ( W `  Z )  /\  ( Z F ( W `  Z
) )  e.  Z
) )
2322simprd 450 . . . . . . . 8  |-  ( ph  ->  ( Z F ( W `  Z ) )  e.  Z )
2422simpld 446 . . . . . . . . . . . . 13  |-  ( ph  ->  Z W ( W `
 Z ) )
254, 12fpwwe2lem2 8433 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Z W ( W `  Z )  <-> 
( ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z ) )  /\  ( ( W `
 Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `  Z
) " { b } )  /  v ]. ( v F ( ( W `  Z
)  i^i  ( v  X.  v ) ) )  =  b ) ) ) )
2624, 25mpbid 202 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z ) )  /\  ( ( W `
 Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `  Z
) " { b } )  /  v ]. ( v F ( ( W `  Z
)  i^i  ( v  X.  v ) ) )  =  b ) ) )
2726simpld 446 . . . . . . . . . . 11  |-  ( ph  ->  ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z ) ) )
2827simpld 446 . . . . . . . . . 10  |-  ( ph  ->  Z  C_  A )
2912, 28ssexd 4284 . . . . . . . . 9  |-  ( ph  ->  Z  e.  _V )
30 sseq1 3305 . . . . . . . . . . . . . 14  |-  ( a  =  Z  ->  (
a  C_  A  <->  Z  C_  A
) )
31 id 20 . . . . . . . . . . . . . . . 16  |-  ( a  =  Z  ->  a  =  Z )
3231, 31xpeq12d 4836 . . . . . . . . . . . . . . 15  |-  ( a  =  Z  ->  (
a  X.  a )  =  ( Z  X.  Z ) )
3332sseq2d 3312 . . . . . . . . . . . . . 14  |-  ( a  =  Z  ->  (
( W `  Z
)  C_  ( a  X.  a )  <->  ( W `  Z )  C_  ( Z  X.  Z ) ) )
34 weeq2 4505 . . . . . . . . . . . . . 14  |-  ( a  =  Z  ->  (
( W `  Z
)  We  a  <->  ( W `  Z )  We  Z
) )
3530, 33, 343anbi123d 1254 . . . . . . . . . . . . 13  |-  ( a  =  Z  ->  (
( a  C_  A  /\  ( W `  Z
)  C_  ( a  X.  a )  /\  ( W `  Z )  We  a )  <->  ( Z  C_  A  /\  ( W `
 Z )  C_  ( Z  X.  Z
)  /\  ( W `  Z )  We  Z
) ) )
3635anbi2d 685 . . . . . . . . . . . 12  |-  ( a  =  Z  ->  (
( ph  /\  (
a  C_  A  /\  ( W `  Z ) 
C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) )  <->  ( ph  /\  ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) ) ) )
37 id 20 . . . . . . . . . . . . . . . 16  |-  ( ( Z  C_  A  /\  ( W `  Z ) 
C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z )  ->  ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z
)  /\  ( W `  Z )  We  Z
) )
38373expa 1153 . . . . . . . . . . . . . . 15  |-  ( ( ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z ) )  /\  ( W `  Z )  We  Z )  -> 
( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) )
3938adantrr 698 . . . . . . . . . . . . . 14  |-  ( ( ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z ) )  /\  ( ( W `  Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `
 Z ) " { b } )  /  v ]. (
v F ( ( W `  Z )  i^i  ( v  X.  v ) ) )  =  b ) )  ->  ( Z  C_  A  /\  ( W `  Z )  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) )
4026, 39syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( Z  C_  A  /\  ( W `  Z
)  C_  ( Z  X.  Z )  /\  ( W `  Z )  We  Z ) )
4140pm4.71i 614 . . . . . . . . . . . 12  |-  ( ph  <->  (
ph  /\  ( Z  C_  A  /\  ( W `
 Z )  C_  ( Z  X.  Z
)  /\  ( W `  Z )  We  Z
) ) )
4236, 41syl6bbr 255 . . . . . . . . . . 11  |-  ( a  =  Z  ->  (
( ph  /\  (
a  C_  A  /\  ( W `  Z ) 
C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) )  <->  ph ) )
43 oveq1 6020 . . . . . . . . . . . . 13  |-  ( a  =  Z  ->  (
a F ( W `
 Z ) )  =  ( Z F ( W `  Z
) ) )
4443, 31eleq12d 2448 . . . . . . . . . . . 12  |-  ( a  =  Z  ->  (
( a F ( W `  Z ) )  e.  a  <->  ( Z F ( W `  Z ) )  e.  Z ) )
45 breq1 4149 . . . . . . . . . . . 12  |-  ( a  =  Z  ->  (
a  ~<  om  <->  Z  ~<  om )
)
4644, 45imbi12d 312 . . . . . . . . . . 11  |-  ( a  =  Z  ->  (
( ( a F ( W `  Z
) )  e.  a  ->  a  ~<  om )  <->  ( ( Z F ( W `  Z ) )  e.  Z  ->  Z  ~<  om ) ) )
4742, 46imbi12d 312 . . . . . . . . . 10  |-  ( a  =  Z  ->  (
( ( ph  /\  ( a  C_  A  /\  ( W `  Z
)  C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) )  -> 
( ( a F ( W `  Z
) )  e.  a  ->  a  ~<  om )
)  <->  ( ph  ->  ( ( Z F ( W `  Z ) )  e.  Z  ->  Z  ~<  om ) ) ) )
48 fvex 5675 . . . . . . . . . . 11  |-  ( W `
 Z )  e. 
_V
49 sseq1 3305 . . . . . . . . . . . . . 14  |-  ( s  =  ( W `  Z )  ->  (
s  C_  ( a  X.  a )  <->  ( W `  Z )  C_  (
a  X.  a ) ) )
50 weeq1 4504 . . . . . . . . . . . . . 14  |-  ( s  =  ( W `  Z )  ->  (
s  We  a  <->  ( W `  Z )  We  a
) )
5149, 503anbi23d 1257 . . . . . . . . . . . . 13  |-  ( s  =  ( W `  Z )  ->  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  <->  ( a  C_  A  /\  ( W `  Z ) 
C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) ) )
5251anbi2d 685 . . . . . . . . . . . 12  |-  ( s  =  ( W `  Z )  ->  (
( ph  /\  (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a ) )  <->  ( ph  /\  ( a  C_  A  /\  ( W `  Z
)  C_  ( a  X.  a )  /\  ( W `  Z )  We  a ) ) ) )
53 oveq2 6021 . . . . . . . . . . . . . 14  |-  ( s  =  ( W `  Z )  ->  (
a F s )  =  ( a F ( W `  Z
) ) )
5453eleq1d 2446 . . . . . . . . . . . . 13  |-  ( s  =  ( W `  Z )  ->  (
( a F s )  e.  a  <->  ( a F ( W `  Z ) )  e.  a ) )
5554imbi1d 309 . . . . . . . . . . . 12  |-  ( s  =  ( W `  Z )  ->  (
( ( a F s )  e.  a  ->  a  ~<  om )  <->  ( ( a F ( W `  Z ) )  e.  a  -> 
a  ~<  om ) ) )
5652, 55imbi12d 312 . . . . . . . . . . 11  |-  ( s  =  ( W `  Z )  ->  (
( ( ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )
)  ->  ( (
a F s )  e.  a  ->  a  ~<  om ) )  <->  ( ( ph  /\  ( a  C_  A  /\  ( W `  Z )  C_  (
a  X.  a )  /\  ( W `  Z )  We  a
) )  ->  (
( a F ( W `  Z ) )  e.  a  -> 
a  ~<  om ) ) ) )
57 omelon 7527 . . . . . . . . . . . . . . 15  |-  om  e.  On
58 onenon 7762 . . . . . . . . . . . . . . 15  |-  ( om  e.  On  ->  om  e.  dom  card )
5957, 58ax-mp 8 . . . . . . . . . . . . . 14  |-  om  e.  dom  card
60 simpr3 965 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
s  We  a )
61 19.8a 1754 . . . . . . . . . . . . . . . 16  |-  ( s  We  a  ->  E. s 
s  We  a )
6260, 61syl 16 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  ->  E. s  s  We  a )
63 ween 7842 . . . . . . . . . . . . . . 15  |-  ( a  e.  dom  card  <->  E. s 
s  We  a )
6462, 63sylibr 204 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
a  e.  dom  card )
65 domtri2 7802 . . . . . . . . . . . . . 14  |-  ( ( om  e.  dom  card  /\  a  e.  dom  card )  ->  ( om  ~<_  a  <->  -.  a  ~<  om ) )
6659, 64, 65sylancr 645 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  <->  -.  a  ~<  om ) )
67 nfv 1626 . . . . . . . . . . . . . . . . 17  |-  F/ r ( ph  /\  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )
68 nfcv 2516 . . . . . . . . . . . . . . . . . . 19  |-  F/_ r
a
69 nfmpt22 6073 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ r
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
7018, 69nfcxfr 2513 . . . . . . . . . . . . . . . . . . 19  |-  F/_ r F
71 nfcv 2516 . . . . . . . . . . . . . . . . . . 19  |-  F/_ r
s
7268, 70, 71nfov 6036 . . . . . . . . . . . . . . . . . 18  |-  F/_ r
( a F s )
7372nfel1 2526 . . . . . . . . . . . . . . . . 17  |-  F/ r ( a F s )  e.  ( A 
\  a )
7467, 73nfim 1822 . . . . . . . . . . . . . . . 16  |-  F/ r ( ( ph  /\  ( ( a  C_  A  /\  s  C_  (
a  X.  a )  /\  s  We  a
)  /\  om  ~<_  a ) )  ->  ( a F s )  e.  ( A  \  a
) )
75 sseq1 3305 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  s  ->  (
r  C_  ( a  X.  a )  <->  s  C_  ( a  X.  a
) ) )
76 weeq1 4504 . . . . . . . . . . . . . . . . . . . 20  |-  ( r  =  s  ->  (
r  We  a  <->  s  We  a ) )
7775, 763anbi23d 1257 . . . . . . . . . . . . . . . . . . 19  |-  ( r  =  s  ->  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  <->  ( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a ) ) )
7877anbi1d 686 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  s  ->  (
( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a )  <-> 
( ( a  C_  A  /\  s  C_  (
a  X.  a )  /\  s  We  a
)  /\  om  ~<_  a ) ) )
7978anbi2d 685 . . . . . . . . . . . . . . . . 17  |-  ( r  =  s  ->  (
( ph  /\  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )  <-> 
( ph  /\  (
( a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) ) ) )
80 oveq2 6021 . . . . . . . . . . . . . . . . . 18  |-  ( r  =  s  ->  (
a F r )  =  ( a F s ) )
8180eleq1d 2446 . . . . . . . . . . . . . . . . 17  |-  ( r  =  s  ->  (
( a F r )  e.  ( A 
\  a )  <->  ( a F s )  e.  ( A  \  a
) ) )
8279, 81imbi12d 312 . . . . . . . . . . . . . . . 16  |-  ( r  =  s  ->  (
( ( ph  /\  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) )  ->  ( a F r )  e.  ( A  \  a
) )  <->  ( ( ph  /\  ( ( a 
C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  ( A 
\  a ) ) ) )
83 nfv 1626 . . . . . . . . . . . . . . . . . 18  |-  F/ x
( ph  /\  (
( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )
84 nfcv 2516 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x
a
85 nfmpt21 6072 . . . . . . . . . . . . . . . . . . . . 21  |-  F/_ x
( x  e.  _V ,  r  e.  _V  |->  if ( x  e.  Fin ,  ( H `  ( card `  x ) ) ,  ( D `  |^| { z  e.  om  |  -.  ( D `  z )  e.  x } ) ) )
8618, 85nfcxfr 2513 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x F
87 nfcv 2516 . . . . . . . . . . . . . . . . . . . 20  |-  F/_ x
r
8884, 86, 87nfov 6036 . . . . . . . . . . . . . . . . . . 19  |-  F/_ x
( a F r )
8988nfel1 2526 . . . . . . . . . . . . . . . . . 18  |-  F/ x
( a F r )  e.  ( A 
\  a )
9083, 89nfim 1822 . . . . . . . . . . . . . . . . 17  |-  F/ x
( ( ph  /\  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) )  ->  ( a F r )  e.  ( A  \  a
) )
91 sseq1 3305 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  ->  (
x  C_  A  <->  a  C_  A ) )
92 xpeq12 4830 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( x  =  a  /\  x  =  a )  ->  ( x  X.  x
)  =  ( a  X.  a ) )
9392anidms 627 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( x  =  a  ->  (
x  X.  x )  =  ( a  X.  a ) )
9493sseq2d 3312 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  ->  (
r  C_  ( x  X.  x )  <->  r  C_  ( a  X.  a
) ) )
95 weeq2 4505 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  a  ->  (
r  We  x  <->  r  We  a ) )
9691, 94, 953anbi123d 1254 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  a  ->  (
( x  C_  A  /\  r  C_  ( x  X.  x )  /\  r  We  x )  <->  ( a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a ) ) )
97 breq2 4150 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  a  ->  ( om 
~<_  x  <->  om  ~<_  a ) )
9896, 97anbi12d 692 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  a  ->  (
( ( x  C_  A  /\  r  C_  (
x  X.  x )  /\  r  We  x
)  /\  om  ~<_  x )  <-> 
( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) ) )
9915, 98syl5bb 249 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  a  ->  ( ps 
<->  ( ( a  C_  A  /\  r  C_  (
a  X.  a )  /\  r  We  a
)  /\  om  ~<_  a ) ) )
10099anbi2d 685 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  a  ->  (
( ph  /\  ps )  <->  (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) ) ) )
101 oveq1 6020 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  a  ->  (
x F r )  =  ( a F r ) )
102 difeq2 3395 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  a  ->  ( A  \  x )  =  ( A  \  a
) )
103101, 102eleq12d 2448 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  a  ->  (
( x F r )  e.  ( A 
\  x )  <->  ( a F r )  e.  ( A  \  a
) ) )
104100, 103imbi12d 312 . . . . . . . . . . . . . . . . 17  |-  ( x  =  a  ->  (
( ( ph  /\  ps )  ->  ( x F r )  e.  ( A  \  x
) )  <->  ( ( ph  /\  ( ( a 
C_  A  /\  r  C_  ( a  X.  a
)  /\  r  We  a )  /\  om  ~<_  a ) )  -> 
( a F r )  e.  ( A 
\  a ) ) ) )
1055, 13, 14, 15, 16, 17, 18pwfseqlem3 8461 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ps )  ->  ( x F r )  e.  ( A 
\  x ) )
10690, 104, 105chvar 2012 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( (
a  C_  A  /\  r  C_  ( a  X.  a )  /\  r  We  a )  /\  om  ~<_  a ) )  -> 
( a F r )  e.  ( A 
\  a ) )
10774, 82, 106chvar 2012 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  -> 
( a F s )  e.  ( A 
\  a ) )
108107eldifbd 3269 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( (
a  C_  A  /\  s  C_  ( a  X.  a )  /\  s  We  a )  /\  om  ~<_  a ) )  ->  -.  ( a F s )  e.  a )
109108expr 599 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( om  ~<_  a  ->  -.  ( a F s )  e.  a ) )
11066, 109sylbird 227 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( -.  a  ~<  om  ->  -.  ( a F s )  e.  a ) )
111110con4d 99 . . . . . . . . . . 11  |-  ( (
ph  /\  ( a  C_  A  /\  s  C_  ( a  X.  a
)  /\  s  We  a ) )  -> 
( ( a F s )  e.  a  ->  a  ~<  om )
)
11248, 56, 111vtocl 2942 . . . . . . . . . 10  |-  ( (
ph  /\  ( a  C_  A  /\  ( W `
 Z )  C_  ( a  X.  a
)  /\  ( W `  Z )  We  a
) )  ->  (
( a F ( W `  Z ) )  e.  a  -> 
a  ~<  om ) )
11347, 112vtoclg 2947 . . . . . . . . 9  |-  ( Z  e.  _V  ->  ( ph  ->  ( ( Z F ( W `  Z ) )  e.  Z  ->  Z  ~<  om ) ) )
11429, 113mpcom 34 . . . . . . . 8  |-  ( ph  ->  ( ( Z F ( W `  Z
) )  e.  Z  ->  Z  ~<  om )
)
11523, 114mpd 15 . . . . . . 7  |-  ( ph  ->  Z  ~<  om )
116 isfinite 7533 . . . . . . 7  |-  ( Z  e.  Fin  <->  Z  ~<  om )
117115, 116sylibr 204 . . . . . 6  |-  ( ph  ->  Z  e.  Fin )
1185, 13, 14, 15, 16, 17, 18pwfseqlem2 8460 . . . . . 6  |-  ( ( Z  e.  Fin  /\  ( W `  Z )  e.  _V )  -> 
( Z F ( W `  Z ) )  =  ( H `
 ( card `  Z
) ) )
119117, 48, 118sylancl 644 . . . . 5  |-  ( ph  ->  ( Z F ( W `  Z ) )  =  ( H `
 ( card `  Z
) ) )
120119, 23eqeltrrd 2455 . . . 4  |-  ( ph  ->  ( H `  ( card `  Z ) )  e.  Z )
1214, 12, 24fpwwe2lem3 8434 . . . . . . . . . 10  |-  ( (
ph  /\  ( H `  ( card `  Z
) )  e.  Z
)  ->  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) F ( ( W `  Z
)  i^i  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  X.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) ) ) )  =  ( H `
 ( card `  Z
) ) )
122120, 121mpdan 650 . . . . . . . . 9  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) F ( ( W `
 Z )  i^i  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  X.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )  =  ( H `  ( card `  Z ) ) )
123 cnvimass 5157 . . . . . . . . . . . 12  |-  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  C_  dom  ( W `
 Z )
12427simprd 450 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( W `  Z
)  C_  ( Z  X.  Z ) )
125 dmss 5002 . . . . . . . . . . . . . 14  |-  ( ( W `  Z ) 
C_  ( Z  X.  Z )  ->  dom  ( W `  Z ) 
C_  dom  ( Z  X.  Z ) )
126124, 125syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  dom  ( W `  Z )  C_  dom  ( Z  X.  Z
) )
127 dmxpss 5233 . . . . . . . . . . . . 13  |-  dom  ( Z  X.  Z )  C_  Z
128126, 127syl6ss 3296 . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( W `  Z )  C_  Z
)
129123, 128syl5ss 3295 . . . . . . . . . . 11  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  C_  Z )
130 ssfi 7258 . . . . . . . . . . 11  |-  ( ( Z  e.  Fin  /\  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C_  Z
)  ->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  e.  Fin )
131117, 129, 130syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
Fin )
13248inex1 4278 . . . . . . . . . 10  |-  ( ( W `  Z )  i^i  ( ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  X.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } ) ) )  e. 
_V
1335, 13, 14, 15, 16, 17, 18pwfseqlem2 8460 . . . . . . . . . 10  |-  ( ( ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
Fin  /\  ( ( W `  Z )  i^i  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  X.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) )  e.  _V )  ->  ( ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } ) F ( ( W `  Z )  i^i  ( ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  X.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } ) ) ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
134131, 132, 133sylancl 644 . . . . . . . . 9  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) F ( ( W `
 Z )  i^i  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } )  X.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
135122, 134eqtr3d 2414 . . . . . . . 8  |-  ( ph  ->  ( H `  ( card `  Z ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
136 f1of1 5606 . . . . . . . . . 10  |-  ( H : om -1-1-onto-> X  ->  H : om
-1-1-> X )
13714, 136syl 16 . . . . . . . . 9  |-  ( ph  ->  H : om -1-1-> X
)
138 ficardom 7774 . . . . . . . . . 10  |-  ( Z  e.  Fin  ->  ( card `  Z )  e. 
om )
139117, 138syl 16 . . . . . . . . 9  |-  ( ph  ->  ( card `  Z
)  e.  om )
140 ficardom 7774 . . . . . . . . . 10  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  e.  Fin  ->  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  e. 
om )
141131, 140syl 16 . . . . . . . . 9  |-  ( ph  ->  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  e. 
om )
142 f1fveq 5940 . . . . . . . . 9  |-  ( ( H : om -1-1-> X  /\  ( ( card `  Z
)  e.  om  /\  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) )  e.  om )
)  ->  ( ( H `  ( card `  Z ) )  =  ( H `  ( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) )  <->  ( card `  Z )  =  (
card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) ) ) )
143137, 139, 141, 142syl12anc 1182 . . . . . . . 8  |-  ( ph  ->  ( ( H `  ( card `  Z )
)  =  ( H `
 ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) ) )  <-> 
( card `  Z )  =  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) ) ) )
144135, 143mpbid 202 . . . . . . 7  |-  ( ph  ->  ( card `  Z
)  =  ( card `  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } ) ) )
145144eqcomd 2385 . . . . . 6  |-  ( ph  ->  ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  =  ( card `  Z
) )
146 finnum 7761 . . . . . . . 8  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  e.  Fin  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
dom  card )
147131, 146syl 16 . . . . . . 7  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
dom  card )
148 finnum 7761 . . . . . . . 8  |-  ( Z  e.  Fin  ->  Z  e.  dom  card )
149117, 148syl 16 . . . . . . 7  |-  ( ph  ->  Z  e.  dom  card )
150 carden2 7800 . . . . . . 7  |-  ( ( ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  e. 
dom  card  /\  Z  e.  dom  card )  ->  (
( card `  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) )  =  ( card `  Z )  <->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~~  Z ) )
151147, 149, 150syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( card `  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } ) )  =  ( card `  Z
)  <->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~~  Z ) )
152145, 151mpbid 202 . . . . 5  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  ~~  Z )
153 dfpss2 3368 . . . . . . . 8  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C.  Z  <->  ( ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  C_  Z  /\  -.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  =  Z ) )
154153baib 872 . . . . . . 7  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C_  Z  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
C.  Z  <->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  =  Z ) )
155129, 154syl 16 . . . . . 6  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
C.  Z  <->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  =  Z ) )
156 php3 7222 . . . . . . . . 9  |-  ( ( Z  e.  Fin  /\  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C.  Z
)  ->  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~<  Z )
157 sdomnen 7065 . . . . . . . . 9  |-  ( ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  ~<  Z  ->  -.  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  ~~  Z )
158156, 157syl 16 . . . . . . . 8  |-  ( ( Z  e.  Fin  /\  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  C.  Z
)  ->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  ~~  Z
)
159158ex 424 . . . . . . 7  |-  ( Z  e.  Fin  ->  (
( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  C.  Z  ->  -.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
~~  Z ) )
160117, 159syl 16 . . . . . 6  |-  ( ph  ->  ( ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) 
C.  Z  ->  -.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  ~~  Z
) )
161155, 160sylbird 227 . . . . 5  |-  ( ph  ->  ( -.  ( `' ( W `  Z
) " { ( H `  ( card `  Z ) ) } )  =  Z  ->  -.  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  ~~  Z ) )
162152, 161mt4d 132 . . . 4  |-  ( ph  ->  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  =  Z )
163120, 162eleqtrrd 2457 . . 3  |-  ( ph  ->  ( H `  ( card `  Z ) )  e.  ( `' ( W `  Z )
" { ( H `
 ( card `  Z
) ) } ) )
164 fvex 5675 . . . 4  |-  ( H `
 ( card `  Z
) )  e.  _V
165164eliniseg 5166 . . . 4  |-  ( ( H `  ( card `  Z ) )  e. 
_V  ->  ( ( H `
 ( card `  Z
) )  e.  ( `' ( W `  Z ) " {
( H `  ( card `  Z ) ) } )  <->  ( H `  ( card `  Z
) ) ( W `
 Z ) ( H `  ( card `  Z ) ) ) )
166164, 165ax-mp 8 . . 3  |-  ( ( H `  ( card `  Z ) )  e.  ( `' ( W `
 Z ) " { ( H `  ( card `  Z )
) } )  <->  ( H `  ( card `  Z
) ) ( W `
 Z ) ( H `  ( card `  Z ) ) )
167163, 166sylib 189 . 2  |-  ( ph  ->  ( H `  ( card `  Z ) ) ( W `  Z
) ( H `  ( card `  Z )
) )
16826simprd 450 . . . . 5  |-  ( ph  ->  ( ( W `  Z )  We  Z  /\  A. b  e.  Z  [. ( `' ( W `
 Z ) " { b } )  /  v ]. (
v F ( ( W `  Z )  i^i  ( v  X.  v ) ) )  =  b ) )
169168simpld 446 . . . 4  |-  ( ph  ->  ( W `  Z
)  We  Z )
170 weso 4507 . . . 4  |-  ( ( W `  Z )  We  Z  ->  ( W `  Z )  Or  Z )
171169, 170syl 16 . . 3  |-  ( ph  ->  ( W `  Z
)  Or  Z )
172 sonr 4458 . . 3  |-  ( ( ( W `  Z
)  Or  Z  /\  ( H `  ( card `  Z ) )  e.  Z )  ->  -.  ( H `  ( card `  Z ) ) ( W `  Z ) ( H `  ( card `  Z ) ) )
173171, 120, 172syl2anc 643 . 2  |-  ( ph  ->  -.  ( H `  ( card `  Z )
) ( W `  Z ) ( H `
 ( card `  Z
) ) )
174167, 173pm2.65i 167 1  |-  -.  ph
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1547    = wceq 1649    e. wcel 1717   A.wral 2642   {crab 2646   _Vcvv 2892   [.wsbc 3097    \ cdif 3253    i^i cin 3255    C_ wss 3256    C. wpss 3257   ifcif 3675   ~Pcpw 3735   {csn 3750   U.cuni 3950   |^|cint 3985   U_ciun 4028   class class class wbr 4146   {copab 4199    Or wor 4436    We wwe 4474   Oncon0 4515   omcom 4778    X. cxp 4809   `'ccnv 4810   dom cdm 4811   ran crn 4812   "cima 4814   -1-1->wf1 5384   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015    ^m cmap 6947    ~~ cen 7035    ~<_ cdom 7036    ~< csdm 7037   Fincfn 7038   cardccrd 7748
This theorem is referenced by:  pwfseqlem5  8464
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-oi 7405  df-card 7752
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