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Theorem infmap2 8090
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8443 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 6081 . . 3  |-  ( B  =  (/)  ->  ( A  ^m  B )  =  ( A  ^m  (/) ) )
2 breq2 4208 . . . . 5  |-  ( B  =  (/)  ->  ( x 
~~  B  <->  x  ~~  (/) ) )
32anbi2d 685 . . . 4  |-  ( B  =  (/)  ->  ( ( x  C_  A  /\  x  ~~  B )  <->  ( x  C_  A  /\  x  ~~  (/) ) ) )
43abbidv 2549 . . 3  |-  ( B  =  (/)  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  =  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
51, 4breq12d 4217 . 2  |-  ( B  =  (/)  ->  ( ( A  ^m  B ) 
~~  { x  |  ( x  C_  A  /\  x  ~~  B ) }  <->  ( A  ^m  (/) )  ~~  { x  |  ( x  C_  A  /\  x  ~~  (/) ) } ) )
6 simpl2 961 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  ~<_  A )
7 reldom 7107 . . . . . . . . . . 11  |-  Rel  ~<_
87brrelexi 4910 . . . . . . . . . 10  |-  ( B  ~<_  A  ->  B  e.  _V )
96, 8syl 16 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  e.  _V )
107brrelex2i 4911 . . . . . . . . . 10  |-  ( B  ~<_  A  ->  A  e.  _V )
116, 10syl 16 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  e.  _V )
12 xpcomeng 7192 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B  X.  A
)  ~~  ( A  X.  B ) )
139, 11, 12syl2anc 643 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( B  X.  A )  ~~  ( A  X.  B
) )
14 simpl3 962 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  e. 
dom  card )
15 simpr 448 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  =/=  (/) )
16 mapdom3 7271 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
1711, 9, 15, 16syl3anc 1184 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
18 numdom 7911 . . . . . . . . . 10  |-  ( ( ( A  ^m  B
)  e.  dom  card  /\  A  ~<_  ( A  ^m  B ) )  ->  A  e.  dom  card )
1914, 17, 18syl2anc 643 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  e.  dom  card )
20 simpl1 960 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  om  ~<_  A )
21 infxpabs 8084 . . . . . . . . 9  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
2219, 20, 15, 6, 21syl22anc 1185 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  X.  B )  ~~  A )
23 entr 7151 . . . . . . . 8  |-  ( ( ( B  X.  A
)  ~~  ( A  X.  B )  /\  ( A  X.  B )  ~~  A )  ->  ( B  X.  A )  ~~  A )
2413, 22, 23syl2anc 643 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( B  X.  A )  ~~  A )
25 ssenen 7273 . . . . . . 7  |-  ( ( B  X.  A ) 
~~  A  ->  { x  |  ( x  C_  ( B  X.  A
)  /\  x  ~~  B ) }  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
2624, 25syl 16 . . . . . 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 7106 . . . . . . 7  |-  Rel  ~~
2827brrelexi 4910 . . . . . 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 16 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  ( B  X.  A
)  /\  x  ~~  B ) }  e.  _V )
30 abid2 2552 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) }  =  ( A  ^m  B )
31 elmapi 7030 . . . . . . . 8  |-  ( x  e.  ( A  ^m  B )  ->  x : B --> A )
32 fssxp 5594 . . . . . . . . 9  |-  ( x : B --> A  ->  x  C_  ( B  X.  A ) )
33 ffun 5585 . . . . . . . . . . 11  |-  ( x : B --> A  ->  Fun  x )
34 vex 2951 . . . . . . . . . . . 12  |-  x  e. 
_V
3534fundmen 7172 . . . . . . . . . . 11  |-  ( Fun  x  ->  dom  x  ~~  x )
36 ensym 7148 . . . . . . . . . . 11  |-  ( dom  x  ~~  x  ->  x  ~~  dom  x )
3733, 35, 363syl 19 . . . . . . . . . 10  |-  ( x : B --> A  ->  x  ~~  dom  x )
38 fdm 5587 . . . . . . . . . 10  |-  ( x : B --> A  ->  dom  x  =  B )
3937, 38breqtrd 4228 . . . . . . . . 9  |-  ( x : B --> A  ->  x  ~~  B )
4032, 39jca 519 . . . . . . . 8  |-  ( x : B --> A  -> 
( x  C_  ( B  X.  A )  /\  x  ~~  B ) )
4131, 40syl 16 . . . . . . 7  |-  ( x  e.  ( A  ^m  B )  ->  (
x  C_  ( B  X.  A )  /\  x  ~~  B ) )
4241ss2abi 3407 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) } 
C_  { x  |  ( x  C_  ( B  X.  A )  /\  x  ~~  B ) }
4330, 42eqsstr3i 3371 . . . . 5  |-  ( A  ^m  B )  C_  { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }
44 ssdomg 7145 . . . . 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 1385 . . . 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 7158 . . . 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 643 . . 3  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  ~<_  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
48 ovex 6098 . . . . . . 7  |-  ( A  ^m  B )  e. 
_V
4948mptex 5958 . . . . . 6  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  e.  _V
5049rnex 5125 . . . . 5  |-  ran  (
f  e.  ( A  ^m  B )  |->  ran  f )  e.  _V
51 ensym 7148 . . . . . . . . . . . 12  |-  ( x 
~~  B  ->  B  ~~  x )
5251ad2antll 710 . . . . . . . . . . 11  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  B  ~~  x
)
53 bren 7109 . . . . . . . . . . 11  |-  ( B 
~~  x  <->  E. f 
f : B -1-1-onto-> x )
5452, 53sylib 189 . . . . . . . . . 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 5666 . . . . . . . . . . . . . . . 16  |-  ( f : B -1-1-onto-> x  ->  f : B --> x )
5655adantl 453 . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . 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 5591 . . . . . . . . . . . . . . 15  |-  ( ( f : B --> x  /\  x  C_  A )  -> 
f : B --> A )
5956, 57, 58syl2anc 643 . . . . . . . . . . . . . 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 7023 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^m  B )  <-> 
f : B --> A ) )
6111, 9, 60syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  (
f  e.  ( A  ^m  B )  <->  f : B
--> A ) )
6261ad2antrr 707 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . 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 5673 . . . . . . . . . . . . . . . 16  |-  ( f : B -1-1-onto-> x  ->  f : B -onto-> x )
65 forn 5648 . . . . . . . . . . . . . . . 16  |-  ( f : B -onto-> x  ->  ran  f  =  x
)
6664, 65syl 16 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> x  ->  ran  f  =  x )
6766adantl 453 . . . . . . . . . . . . . 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 2440 . . . . . . . . . . . . 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 519 . . . . . . . . . . . 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 424 . . . . . . . . . . 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 1632 . . . . . . . . . 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 15 . . . . . . . . 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 2703 . . . . . . . . 9  |-  ( E. f  e.  ( A  ^m  B ) x  =  ran  f  <->  E. f
( f  e.  ( A  ^m  B )  /\  x  =  ran  f ) )
7472, 73sylibr 204 . . . . . . . 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 424 . . . . . . 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 3408 . . . . . 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 2435 . . . . . . 7  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  =  ( f  e.  ( A  ^m  B )  |->  ran  f
)
7877rnmpt 5108 . . . . . 6  |-  ran  (
f  e.  ( A  ^m  B )  |->  ran  f )  =  {
x  |  E. f  e.  ( A  ^m  B
) x  =  ran  f }
7976, 78syl6sseqr 3387 . . . . 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 7145 . . . . 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 61 . . . 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 2951 . . . . . . . . 9  |-  f  e. 
_V
8382rnex 5125 . . . . . . . 8  |-  ran  f  e.  _V
8483rgenw 2765 . . . . . . 7  |-  A. f  e.  ( A  ^m  B
) ran  f  e.  _V
8577fnmpt 5563 . . . . . . 7  |-  ( A. f  e.  ( A  ^m  B ) ran  f  e.  _V  ->  ( f  e.  ( A  ^m  B
)  |->  ran  f )  Fn  ( A  ^m  B
) )
8684, 85mp1i 12 . . . . . 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 5651 . . . . . 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 189 . . . . 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 7930 . . . . 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 58 . . . 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 7152 . . . 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 643 . . 3  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  ~<_  ( A  ^m  B ) )
93 sbth 7219 . . 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 643 . 2  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
957brrelex2i 4911 . . . . 5  |-  ( om  ~<_  A  ->  A  e.  _V )
96953ad2ant1 978 . . . 4  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  A  e.  _V )
97 map0e 7043 . . . 4  |-  ( A  e.  _V  ->  ( A  ^m  (/) )  =  1o )
9896, 97syl 16 . . 3  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  (/) )  =  1o )
99 1onn 6874 . . . . . 6  |-  1o  e.  om
10099elexi 2957 . . . . 5  |-  1o  e.  _V
101100enref 7132 . . . 4  |-  1o  ~~  1o
102 df-sn 3812 . . . . 5  |-  { (/) }  =  { x  |  x  =  (/) }
103 df1o2 6728 . . . . 5  |-  1o  =  { (/) }
104 en0 7162 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
105104anbi2i 676 . . . . . . 7  |-  ( ( x  C_  A  /\  x  ~~  (/) )  <->  ( x  C_  A  /\  x  =  (/) ) )
106 0ss 3648 . . . . . . . . 9  |-  (/)  C_  A
107 sseq1 3361 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( x 
C_  A  <->  (/)  C_  A
) )
108106, 107mpbiri 225 . . . . . . . 8  |-  ( x  =  (/)  ->  x  C_  A )
109108pm4.71ri 615 . . . . . . 7  |-  ( x  =  (/)  <->  ( x  C_  A  /\  x  =  (/) ) )
110105, 109bitr4i 244 . . . . . 6  |-  ( ( x  C_  A  /\  x  ~~  (/) )  <->  x  =  (/) )
111110abbii 2547 . . . . 5  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  { x  |  x  =  (/) }
112102, 103, 1113eqtr4ri 2466 . . . 4  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  1o
113101, 112breqtrri 4229 . . 3  |-  1o  ~~  { x  |  ( x 
C_  A  /\  x  ~~  (/) ) }
11498, 113syl6eqbr 4241 . 2  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  (/) )  ~~  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
1155, 94, 114pm2.61ne 2673 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 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   _Vcvv 2948    C_ wss 3312   (/)c0 3620   {csn 3806   class class class wbr 4204    e. cmpt 4258   omcom 4837    X. cxp 4868   dom cdm 4870   ran crn 4871   Fun wfun 5440    Fn wfn 5441   -->wf 5442   -onto->wfo 5444   -1-1-onto->wf1o 5445  (class class class)co 6073   1oc1o 6709    ^m cmap 7010    ~~ cen 7098    ~<_ cdom 7099   cardccrd 7814
This theorem is referenced by:  infmap  8443
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-acn 7821
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