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Theorem infmap2 7728
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8078 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
StepHypRef Expression
1 oveq2 5718 . . 3  |-  ( B  =  (/)  ->  ( A  ^m  B )  =  ( A  ^m  (/) ) )
2 breq2 3924 . . . . 5  |-  ( B  =  (/)  ->  ( x 
~~  B  <->  x  ~~  (/) ) )
32anbi2d 687 . . . 4  |-  ( B  =  (/)  ->  ( ( x  C_  A  /\  x  ~~  B )  <->  ( x  C_  A  /\  x  ~~  (/) ) ) )
43abbidv 2363 . . 3  |-  ( B  =  (/)  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  =  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
51, 4breq12d 3933 . 2  |-  ( B  =  (/)  ->  ( ( A  ^m  B ) 
~~  { x  |  ( x  C_  A  /\  x  ~~  B ) }  <->  ( A  ^m  (/) )  ~~  { x  |  ( x  C_  A  /\  x  ~~  (/) ) } ) )
6 simpl2 964 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  ~<_  A )
7 reldom 6755 . . . . . . . . . . 11  |-  Rel  ~<_
87brrelexi 4636 . . . . . . . . . 10  |-  ( B  ~<_  A  ->  B  e.  _V )
96, 8syl 17 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  e.  _V )
107brrelex2i 4637 . . . . . . . . . 10  |-  ( B  ~<_  A  ->  A  e.  _V )
116, 10syl 17 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  e.  _V )
12 xpcomeng 6839 . . . . . . . . 9  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( B  X.  A
)  ~~  ( A  X.  B ) )
139, 11, 12syl2anc 645 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( B  X.  A )  ~~  ( A  X.  B
) )
14 simpl3 965 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  e. 
dom  card )
15 simpr 449 . . . . . . . . . . 11  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  B  =/=  (/) )
16 mapdom3 6918 . . . . . . . . . . 11  |-  ( ( A  e.  _V  /\  B  e.  _V  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
1711, 9, 15, 16syl3anc 1187 . . . . . . . . . 10  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  ~<_  ( A  ^m  B ) )
18 numdom 7549 . . . . . . . . . 10  |-  ( ( ( A  ^m  B
)  e.  dom  card  /\  A  ~<_  ( A  ^m  B ) )  ->  A  e.  dom  card )
1914, 17, 18syl2anc 645 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  A  e.  dom  card )
20 simpl1 963 . . . . . . . . 9  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  om  ~<_  A )
21 infxpabs 7722 . . . . . . . . 9  |-  ( ( ( A  e.  dom  card  /\  om  ~<_  A )  /\  ( B  =/=  (/)  /\  B  ~<_  A ) )  -> 
( A  X.  B
)  ~~  A )
2219, 20, 15, 6, 21syl22anc 1188 . . . . . . . 8  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  X.  B )  ~~  A )
23 entr 6798 . . . . . . . 8  |-  ( ( ( B  X.  A
)  ~~  ( A  X.  B )  /\  ( A  X.  B )  ~~  A )  ->  ( B  X.  A )  ~~  A )
2413, 22, 23syl2anc 645 . . . . . . 7  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( B  X.  A )  ~~  A )
25 ssenen 6920 . . . . . . 7  |-  ( ( B  X.  A ) 
~~  A  ->  { x  |  ( x  C_  ( B  X.  A
)  /\  x  ~~  B ) }  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
2624, 25syl 17 . . . . . 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 6754 . . . . . . 7  |-  Rel  ~~
2827brrelexi 4636 . . . . . 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 17 . . . . 5  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  ( B  X.  A
)  /\  x  ~~  B ) }  e.  _V )
30 abid2 2366 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) }  =  ( A  ^m  B )
31 elmapi 6678 . . . . . . . 8  |-  ( x  e.  ( A  ^m  B )  ->  x : B --> A )
32 fssxp 5257 . . . . . . . . 9  |-  ( x : B --> A  ->  x  C_  ( B  X.  A ) )
33 ffun 5248 . . . . . . . . . . 11  |-  ( x : B --> A  ->  Fun  x )
34 vex 2730 . . . . . . . . . . . 12  |-  x  e. 
_V
3534fundmen 6819 . . . . . . . . . . 11  |-  ( Fun  x  ->  dom  x  ~~  x )
36 ensym 6796 . . . . . . . . . . 11  |-  ( dom  x  ~~  x  ->  x  ~~  dom  x )
3733, 35, 363syl 20 . . . . . . . . . 10  |-  ( x : B --> A  ->  x  ~~  dom  x )
38 fdm 5250 . . . . . . . . . 10  |-  ( x : B --> A  ->  dom  x  =  B )
3937, 38breqtrd 3944 . . . . . . . . 9  |-  ( x : B --> A  ->  x  ~~  B )
4032, 39jca 520 . . . . . . . 8  |-  ( x : B --> A  -> 
( x  C_  ( B  X.  A )  /\  x  ~~  B ) )
4131, 40syl 17 . . . . . . 7  |-  ( x  e.  ( A  ^m  B )  ->  (
x  C_  ( B  X.  A )  /\  x  ~~  B ) )
4241ss2abi 3166 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) } 
C_  { x  |  ( x  C_  ( B  X.  A )  /\  x  ~~  B ) }
4330, 42eqsstr3i 3130 . . . . 5  |-  ( A  ^m  B )  C_  { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }
44 ssdomg 6793 . . . . 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 1372 . . . 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 6805 . . . 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 645 . . 3  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  ~<_  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
48 ovex 5735 . . . . . . 7  |-  ( A  ^m  B )  e. 
_V
4948mptex 5598 . . . . . 6  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  e.  _V
5049rnex 4849 . . . . 5  |-  ran  ( 
f  e.  ( A  ^m  B )  |->  ran  f )  e.  _V
51 ensym 6796 . . . . . . . . . . . 12  |-  ( x 
~~  B  ->  B  ~~  x )
5251ad2antll 712 . . . . . . . . . . 11  |-  ( ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  /\  (
x  C_  A  /\  x  ~~  B ) )  ->  B  ~~  x
)
53 bren 6757 . . . . . . . . . . 11  |-  ( B 
~~  x  <->  E. f 
f : B -1-1-onto-> x )
5452, 53sylib 190 . . . . . . . . . 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 5329 . . . . . . . . . . . . . . . 16  |-  ( f : B -1-1-onto-> x  ->  f : B --> x )
5655adantl 454 . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . 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 5254 . . . . . . . . . . . . . . 15  |-  ( ( f : B --> x  /\  x  C_  A )  -> 
f : B --> A )
5956, 57, 58syl2anc 645 . . . . . . . . . . . . . 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 6671 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( f  e.  ( A  ^m  B )  <-> 
f : B --> A ) )
6111, 9, 60syl2anc 645 . . . . . . . . . . . . . . 15  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  (
f  e.  ( A  ^m  B )  <->  f : B
--> A ) )
6261ad2antrr 709 . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . 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 5336 . . . . . . . . . . . . . . . 16  |-  ( f : B -1-1-onto-> x  ->  f : B -onto-> x )
65 forn 5311 . . . . . . . . . . . . . . . 16  |-  ( f : B -onto-> x  ->  ran  f  =  x
)
6664, 65syl 17 . . . . . . . . . . . . . . 15  |-  ( f : B -1-1-onto-> x  ->  ran  f  =  x )
6766adantl 454 . . . . . . . . . . . . . 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 2258 . . . . . . . . . . . . 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 520 . . . . . . . . . . . 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 425 . . . . . . . . . . 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 2018 . . . . . . . . . 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 16 . . . . . . . . 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 2514 . . . . . . . . 9  |-  ( E. f  e.  ( A  ^m  B ) x  =  ran  f  <->  E. f
( f  e.  ( A  ^m  B )  /\  x  =  ran  f ) )
7472, 73sylibr 205 . . . . . . . 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 425 . . . . . . 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 3167 . . . . . 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 2253 . . . . . . 7  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  =  ( f  e.  ( A  ^m  B )  |->  ran  f
)
7877rnmpt 4832 . . . . . 6  |-  ran  ( 
f  e.  ( A  ^m  B )  |->  ran  f )  =  {
x  |  E. f  e.  ( A  ^m  B
) x  =  ran  f }
7976, 78syl6sseqr 3146 . . . . 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 6793 . . . . 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 2730 . . . . . . . . 9  |-  f  e. 
_V
8382rnex 4849 . . . . . . . 8  |-  ran  f  e.  _V
8483rgenw 2572 . . . . . . 7  |-  A. f  e.  ( A  ^m  B
) ran  f  e.  _V
8577fnmpt 5227 . . . . . . 7  |-  ( A. f  e.  ( A  ^m  B ) ran  f  e.  _V  ->  ( f  e.  ( A  ^m  B
)  |->  ran  f )  Fn  ( A  ^m  B
) )
8684, 85mp1i 13 . . . . . 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 5314 . . . . . 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 190 . . . . 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 7568 . . . . 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 6799 . . . 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 645 . . 3  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  ~<_  ( A  ^m  B ) )
93 sbth 6866 . . 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 645 . 2  |-  ( ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e. 
dom  card )  /\  B  =/=  (/) )  ->  ( A  ^m  B )  ~~  { x  |  ( x 
C_  A  /\  x  ~~  B ) } )
957brrelex2i 4637 . . . . 5  |-  ( om  ~<_  A  ->  A  e.  _V )
96953ad2ant1 981 . . . 4  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  A  e.  _V )
97 map0e 6691 . . . 4  |-  ( A  e.  _V  ->  ( A  ^m  (/) )  =  1o )
9896, 97syl 17 . . 3  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  (/) )  =  1o )
99 1onn 6523 . . . . . 6  |-  1o  e.  om
10099elexi 2736 . . . . 5  |-  1o  e.  _V
101100enref 6780 . . . 4  |-  1o  ~~  1o
102 df-sn 3550 . . . . 5  |-  { (/) }  =  { x  |  x  =  (/) }
103 df1o2 6377 . . . . 5  |-  1o  =  { (/) }
104 en0 6809 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
105104anbi2i 678 . . . . . . 7  |-  ( ( x  C_  A  /\  x  ~~  (/) )  <->  ( x  C_  A  /\  x  =  (/) ) )
106 0ss 3390 . . . . . . . . 9  |-  (/)  C_  A
107 sseq1 3120 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( x 
C_  A  <->  (/)  C_  A
) )
108106, 107mpbiri 226 . . . . . . . 8  |-  ( x  =  (/)  ->  x  C_  A )
109108pm4.71ri 617 . . . . . . 7  |-  ( x  =  (/)  <->  ( x  C_  A  /\  x  =  (/) ) )
110105, 109bitr4i 245 . . . . . 6  |-  ( ( x  C_  A  /\  x  ~~  (/) )  <->  x  =  (/) )
111110abbii 2361 . . . . 5  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  { x  |  x  =  (/) }
112102, 103, 1113eqtr4ri 2284 . . . 4  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  1o
113101, 112breqtrri 3945 . . 3  |-  1o  ~~  { x  |  ( x 
C_  A  /\  x  ~~  (/) ) }
11498, 113syl6eqbr 3957 . 2  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  (/) )  ~~  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
1155, 94, 114pm2.61ne 2487 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 6    <-> wb 178    /\ wa 360    /\ w3a 939   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2239    =/= wne 2412   A.wral 2509   E.wrex 2510   _Vcvv 2727    C_ wss 3078   (/)c0 3362   {csn 3544   class class class wbr 3920    e. cmpt 3974   omcom 4547    X. cxp 4578   dom cdm 4580   ran crn 4581   Fun wfun 4586    Fn wfn 4587   -->wf 4588   -onto->wfo 4590   -1-1-onto->wf1o 4591  (class class class)co 5710   1oc1o 6358    ^m cmap 6658    ~~ cen 6746    ~<_ cdom 6747   cardccrd 7452
This theorem is referenced by:  infmap  8078
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-card 7456  df-acn 7459
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