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Theorem infmap2 7840
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8194 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infmap2  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  B )  ~~  {
x  |  ( x 
C_  A  /\  x  ~~  B ) } )
Distinct variable groups:    x, A    x, B

Proof of Theorem infmap2
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 oveq2 5828 . . 3  |-  ( B  =  (/)  ->  ( A  ^m  B )  =  ( A  ^m  (/) ) )
2 breq2 4028 . . . . 5  |-  ( B  =  (/)  ->  ( x 
~~  B  <->  x  ~~  (/) ) )
32anbi2d 684 . . . 4  |-  ( B  =  (/)  ->  ( ( x  C_  A  /\  x  ~~  B )  <->  ( x  C_  A  /\  x  ~~  (/) ) ) )
43abbidv 2398 . . 3  |-  ( B  =  (/)  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  =  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
51, 4breq12d 4037 . 2  |-  ( B  =  (/)  ->  ( ( A  ^m  B ) 
~~  { x  |  ( x  C_  A  /\  x  ~~  B ) }  <->  ( A  ^m  (/) )  ~~  { x  |  ( x  C_  A  /\  x  ~~  (/) ) } ) )
6 simpl2 959 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  ~<_  A )
7 reldom 6865 . . . . . . . . . . 11  |-  Rel  ~<_
87brrelexi 4728 . . . . . . . . . 10  |-  ( B  ~<_  A  ->  B  e.  _V )
96, 8syl 15 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  e.  _V )
107brrelex2i 4729 . . . . . . . . . 10  |-  ( B  ~<_  A  ->  A  e.  _V )
116, 10syl 15 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  e.  _V )
12 xpcomeng 6950 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B  X.  A
)  ~~  ( A  X.  B ) )
139, 11, 12syl2anc 642 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( B  X.  A )  ~~  ( A  X.  B
) )
14 simpl3 960 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  e. 
dom  card )
15 simpr 447 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  =/=  (/) )
16 mapdom3 7029 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
1711, 9, 15, 16syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
18 numdom 7661 . . . . . . . . . 10  |-  ( ( ( A  ^m  B
)  e.  dom  card  /\  A  ~<_  ( A  ^m  B ) )  ->  A  e.  dom  card )
1914, 17, 18syl2anc 642 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  e.  dom  card )
20 simpl1 958 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  om  ~<_  A )
21 infxpabs 7834 . . . . . . . . 9  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
2219, 20, 15, 6, 21syl22anc 1183 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  X.  B )  ~~  A )
23 entr 6909 . . . . . . . 8  |-  ( ( ( B  X.  A
)  ~~  ( A  X.  B )  /\  ( A  X.  B )  ~~  A )  ->  ( B  X.  A )  ~~  A )
2413, 22, 23syl2anc 642 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( B  X.  A )  ~~  A )
25 ssenen 7031 . . . . . . 7  |-  ( ( B  X.  A ) 
~~  A  ->  { x  |  ( x  C_  ( B  X.  A
)  /\  x  ~~  B ) }  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
2624, 25syl 15 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  ( B  X.  A
)  /\  x  ~~  B ) }  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
27 relen 6864 . . . . . . 7  |-  Rel  ~~
2827brrelexi 4728 . . . . . 6  |-  ( { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) }  ->  { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }  e.  _V )
2926, 28syl 15 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  ( B  X.  A
)  /\  x  ~~  B ) }  e.  _V )
30 abid2 2401 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) }  =  ( A  ^m  B )
31 elmapi 6788 . . . . . . . 8  |-  ( x  e.  ( A  ^m  B )  ->  x : B --> A )
32 fssxp 5366 . . . . . . . . 9  |-  ( x : B --> A  ->  x  C_  ( B  X.  A ) )
33 ffun 5357 . . . . . . . . . . 11  |-  ( x : B --> A  ->  Fun  x )
34 vex 2792 . . . . . . . . . . . 12  |-  x  e. 
_V
3534fundmen 6930 . . . . . . . . . . 11  |-  ( Fun  x  ->  dom  x  ~~  x )
36 ensym 6906 . . . . . . . . . . 11  |-  (  dom  x  ~~  x  ->  x  ~~  dom  x )
3733, 35, 363syl 18 . . . . . . . . . 10  |-  ( x : B --> A  ->  x  ~~  dom  x )
38 fdm 5359 . . . . . . . . . 10  |-  ( x : B --> A  ->  dom  x  =  B )
3937, 38breqtrd 4048 . . . . . . . . 9  |-  ( x : B --> A  ->  x  ~~  B )
4032, 39jca 518 . . . . . . . 8  |-  ( x : B --> A  -> 
( x  C_  ( B  X.  A )  /\  x  ~~  B ) )
4131, 40syl 15 . . . . . . 7  |-  ( x  e.  ( A  ^m  B )  ->  (
x  C_  ( B  X.  A )  /\  x  ~~  B ) )
4241ss2abi 3246 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) } 
C_  { x  |  ( x  C_  ( B  X.  A )  /\  x  ~~  B ) }
4330, 42eqsstr3i 3210 . . . . 5  |-  ( A  ^m  B )  C_  { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }
44 ssdomg 6903 . . . . 5  |-  ( { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }  e.  _V  ->  ( ( A  ^m  B )  C_  { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }  ->  ( A  ^m  B )  ~<_  { x  |  ( x  C_  ( B  X.  A )  /\  x  ~~  B ) } ) )
4529, 43, 44ee10 1366 . . . 4  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  ~<_  { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) } )
46 domentr 6916 . . . 4  |-  ( ( ( A  ^m  B
)  ~<_  { x  |  ( x  C_  ( B  X.  A )  /\  x  ~~  B ) }  /\  { x  |  ( x  C_  ( B  X.  A )  /\  x  ~~  B ) } 
~~  { x  |  ( x  C_  A  /\  x  ~~  B ) } )  ->  ( A  ^m  B )  ~<_  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
4745, 26, 46syl2anc 642 . . 3  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  ~<_  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
48 ovex 5845 . . . . . . 7  |-  ( A  ^m  B )  e. 
_V
4948mptex 5708 . . . . . 6  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  e.  _V
5049rnex 4941 . . . . 5  |-  ran  ( 
f  e.  ( A  ^m  B )  |->  ran  f )  e.  _V
51 ensym 6906 . . . . . . . . . . . 12  |-  ( x 
~~  B  ->  B  ~~  x )
5251ad2antll 709 . . . . . . . . . . 11  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  B  ~~  x
)
53 bren 6867 . . . . . . . . . . 11  |-  ( B 
~~  x  <->  E. f 
f : B -1-1-onto-> x )
5452, 53sylib 188 . . . . . . . . . 10  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  E. f  f : B -1-1-onto-> x )
55 f1of 5438 . . . . . . . . . . . . . . . 16  |-  ( f : B -1-1-onto-> x  ->  f : B --> x )
5655adantl 452 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  f : B
--> x )
57 simplrl 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  x  C_  A
)
58 fss 5363 . . . . . . . . . . . . . . 15  |-  ( ( f : B --> x  /\  x  C_  A )  -> 
f : B --> A )
5956, 57, 58syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  f : B
--> A )
60 elmapg 6781 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^m  B )  <-> 
f : B --> A ) )
6111, 9, 60syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  (
f  e.  ( A  ^m  B )  <->  f : B
--> A ) )
6261ad2antrr 706 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  ( f  e.  ( A  ^m  B
)  <->  f : B --> A ) )
6359, 62mpbird 223 . . . . . . . . . . . . 13  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  f  e.  ( A  ^m  B ) )
64 f1ofo 5445 . . . . . . . . . . . . . . . 16  |-  ( f : B -1-1-onto-> x  ->  f : B -onto-> x )
65 forn 5420 . . . . . . . . . . . . . . . 16  |-  ( f : B -onto-> x  ->  ran  f  =  x
)
6664, 65syl 15 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> x  ->  ran  f  =  x )
6766adantl 452 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  ran  f  =  x )
6867eqcomd 2289 . . . . . . . . . . . . 13  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  x  =  ran  f )
6963, 68jca 518 . . . . . . . . . . . 12  |-  ( ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  /\  f : B -1-1-onto-> x
)  ->  ( f  e.  ( A  ^m  B
)  /\  x  =  ran  f ) )
7069ex 423 . . . . . . . . . . 11  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  ( f : B -1-1-onto-> x  ->  ( f  e.  ( A  ^m  B )  /\  x  =  ran  f ) ) )
7170eximdv 1608 . . . . . . . . . 10  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  ( E. f 
f : B -1-1-onto-> x  ->  E. f ( f  e.  ( A  ^m  B
)  /\  x  =  ran  f ) ) )
7254, 71mpd 14 . . . . . . . . 9  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  E. f ( f  e.  ( A  ^m  B )  /\  x  =  ran  f ) )
73 df-rex 2550 . . . . . . . . 9  |-  ( E. f  e.  ( A  ^m  B ) x  =  ran  f  <->  E. f
( f  e.  ( A  ^m  B )  /\  x  =  ran  f ) )
7472, 73sylibr 203 . . . . . . . 8  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  E. f  e.  ( A  ^m  B ) x  =  ran  f
)
7574ex 423 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  (
( x  C_  A  /\  x  ~~  B )  ->  E. f  e.  ( A  ^m  B ) x  =  ran  f
) )
7675ss2abdv 3247 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  C_  { x  |  E. f  e.  ( A  ^m  B ) x  =  ran  f } )
77 eqid 2284 . . . . . . 7  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  =  ( f  e.  ( A  ^m  B )  |->  ran  f
)
7877rnmpt 4924 . . . . . 6  |-  ran  ( 
f  e.  ( A  ^m  B )  |->  ran  f )  =  {
x  |  E. f  e.  ( A  ^m  B
) x  =  ran  f }
7976, 78syl6sseqr 3226 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  C_  ran  (  f  e.  ( A  ^m  B )  |->  ran  f ) )
80 ssdomg 6903 . . . . 5  |-  ( ran  (  f  e.  ( A  ^m  B ) 
|->  ran  f )  e. 
_V  ->  ( { x  |  ( x  C_  A  /\  x  ~~  B
) }  C_  ran  (  f  e.  ( A  ^m  B )  |->  ran  f )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  ~<_  ran  ( 
f  e.  ( A  ^m  B )  |->  ran  f ) ) )
8150, 79, 80mpsyl 59 . . . 4  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  ~<_  ran  ( 
f  e.  ( A  ^m  B )  |->  ran  f ) )
82 vex 2792 . . . . . . . . 9  |-  f  e. 
_V
8382rnex 4941 . . . . . . . 8  |-  ran  f  e.  _V
8483rgenw 2611 . . . . . . 7  |-  A. f  e.  ( A  ^m  B
) ran  f  e.  _V
8577fnmpt 5336 . . . . . . 7  |-  ( A. f  e.  ( A  ^m  B ) ran  f  e.  _V  ->  ( f  e.  ( A  ^m  B
)  |->  ran  f )  Fn  ( A  ^m  B
) )
8684, 85mp1i 11 . . . . . 6  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  (
f  e.  ( A  ^m  B )  |->  ran  f )  Fn  ( A  ^m  B ) )
87 dffn4 5423 . . . . . 6  |-  ( ( f  e.  ( A  ^m  B )  |->  ran  f )  Fn  ( A  ^m  B )  <->  ( f  e.  ( A  ^m  B
)  |->  ran  f ) : ( A  ^m  B ) -onto-> ran  ( 
f  e.  ( A  ^m  B )  |->  ran  f ) )
8886, 87sylib 188 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  (
f  e.  ( A  ^m  B )  |->  ran  f ) : ( A  ^m  B )
-onto->
ran  (  f  e.  ( A  ^m  B
)  |->  ran  f )
)
89 fodomnum 7680 . . . . 5  |-  ( ( A  ^m  B )  e.  dom  card  ->  ( ( f  e.  ( A  ^m  B ) 
|->  ran  f ) : ( A  ^m  B
) -onto-> ran  (  f  e.  ( A  ^m  B
)  |->  ran  f )  ->  ran  (  f  e.  ( A  ^m  B
)  |->  ran  f )  ~<_  ( A  ^m  B ) ) )
9014, 88, 89sylc 56 . . . 4  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ran  (  f  e.  ( A  ^m  B )  |->  ran  f )  ~<_  ( A  ^m  B ) )
91 domtr 6910 . . . 4  |-  ( ( { x  |  ( x  C_  A  /\  x  ~~  B ) }  ~<_  ran  (  f  e.  ( A  ^m  B
)  |->  ran  f )  /\  ran  (  f  e.  ( A  ^m  B
)  |->  ran  f )  ~<_  ( A  ^m  B ) )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  ~<_  ( A  ^m  B ) )
9281, 90, 91syl2anc 642 . . 3  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  ~<_  ( A  ^m  B ) )
93 sbth 6977 . . 3  |-  ( ( ( A  ^m  B
)  ~<_  { x  |  ( x  C_  A  /\  x  ~~  B ) }  /\  { x  |  ( x  C_  A  /\  x  ~~  B
) }  ~<_  ( A  ^m  B ) )  ->  ( A  ^m  B )  ~~  {
x  |  ( x 
C_  A  /\  x  ~~  B ) } )
9447, 92, 93syl2anc 642 . 2  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
957brrelex2i 4729 . . . . 5  |-  ( om  ~<_  A  ->  A  e.  _V )
96953ad2ant1 976 . . . 4  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  A  e.  _V )
97 map0e 6801 . . . 4  |-  ( A  e.  _V  ->  ( A  ^m  (/) )  =  1o )
9896, 97syl 15 . . 3  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  (/) )  =  1o )
99 1onn 6633 . . . . . 6  |-  1o  e.  om
10099elexi 2798 . . . . 5  |-  1o  e.  _V
101100enref 6890 . . . 4  |-  1o  ~~  1o
102 df-sn 3647 . . . . 5  |-  { (/) }  =  { x  |  x  =  (/) }
103 df1o2 6487 . . . . 5  |-  1o  =  { (/) }
104 en0 6920 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
105104anbi2i 675 . . . . . . 7  |-  ( ( x  C_  A  /\  x  ~~  (/) )  <->  ( x  C_  A  /\  x  =  (/) ) )
106 0ss 3484 . . . . . . . . 9  |-  (/)  C_  A
107 sseq1 3200 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( x 
C_  A  <->  (/)  C_  A
) )
108106, 107mpbiri 224 . . . . . . . 8  |-  ( x  =  (/)  ->  x  C_  A )
109108pm4.71ri 614 . . . . . . 7  |-  ( x  =  (/)  <->  ( x  C_  A  /\  x  =  (/) ) )
110105, 109bitr4i 243 . . . . . 6  |-  ( ( x  C_  A  /\  x  ~~  (/) )  <->  x  =  (/) )
111110abbii 2396 . . . . 5  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  { x  |  x  =  (/) }
112102, 103, 1113eqtr4ri 2315 . . . 4  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  1o
113101, 112breqtrri 4049 . . 3  |-  1o  ~~  { x  |  ( x 
C_  A  /\  x  ~~  (/) ) }
11498, 113syl6eqbr 4061 . 2  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  (/) )  ~~  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
1155, 94, 114pm2.61ne 2522 1  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  B )  ~~  {
x  |  ( x 
C_  A  /\  x  ~~  B ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1685   {cab 2270    =/= wne 2447   A.wral 2544   E.wrex 2545   _Vcvv 2789    C_ wss 3153   (/)c0 3456   {csn 3641   class class class wbr 4024    e. cmpt 4078   omcom 4655    X. cxp 4686    dom cdm 4688   ran crn 4689   Fun wfun 5215    Fn wfn 5216   -->wf 5217   -onto->wfo 5219   -1-1-onto->wf1o 5220  (class class class)co 5820   1oc1o 6468    ^m cmap 6768    ~~ cen 6856    ~<_ cdom 6857   cardccrd 7564
This theorem is referenced by:  infmap  8194
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-inf2 7338
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 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-card 7568  df-acn 7571
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