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Theorem infmap2 8024
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 8377 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 6021 . . 3  |-  ( B  =  (/)  ->  ( A  ^m  B )  =  ( A  ^m  (/) ) )
2 breq2 4150 . . . . 5  |-  ( B  =  (/)  ->  ( x 
~~  B  <->  x  ~~  (/) ) )
32anbi2d 685 . . . 4  |-  ( B  =  (/)  ->  ( ( x  C_  A  /\  x  ~~  B )  <->  ( x  C_  A  /\  x  ~~  (/) ) ) )
43abbidv 2494 . . 3  |-  ( B  =  (/)  ->  { x  |  ( x  C_  A  /\  x  ~~  B
) }  =  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
51, 4breq12d 4159 . 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 7044 . . . . . . . . . . 11  |-  Rel  ~<_
87brrelexi 4851 . . . . . . . . . 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 4852 . . . . . . . . . 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 7129 . . . . . . . . 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 7208 . . . . . . . . . . 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 7845 . . . . . . . . . 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 8018 . . . . . . . . 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 7088 . . . . . . . 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 7210 . . . . . . 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 7043 . . . . . . 7  |-  Rel  ~~
2827brrelexi 4851 . . . . . 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 2497 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) }  =  ( A  ^m  B )
31 elmapi 6967 . . . . . . . 8  |-  ( x  e.  ( A  ^m  B )  ->  x : B --> A )
32 fssxp 5535 . . . . . . . . 9  |-  ( x : B --> A  ->  x  C_  ( B  X.  A ) )
33 ffun 5526 . . . . . . . . . . 11  |-  ( x : B --> A  ->  Fun  x )
34 vex 2895 . . . . . . . . . . . 12  |-  x  e. 
_V
3534fundmen 7109 . . . . . . . . . . 11  |-  ( Fun  x  ->  dom  x  ~~  x )
36 ensym 7085 . . . . . . . . . . 11  |-  ( dom  x  ~~  x  ->  x  ~~  dom  x )
3733, 35, 363syl 19 . . . . . . . . . 10  |-  ( x : B --> A  ->  x  ~~  dom  x )
38 fdm 5528 . . . . . . . . . 10  |-  ( x : B --> A  ->  dom  x  =  B )
3937, 38breqtrd 4170 . . . . . . . . 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 3351 . . . . . 6  |-  { x  |  x  e.  ( A  ^m  B ) } 
C_  { x  |  ( x  C_  ( B  X.  A )  /\  x  ~~  B ) }
4330, 42eqsstr3i 3315 . . . . 5  |-  ( A  ^m  B )  C_  { x  |  ( x 
C_  ( B  X.  A )  /\  x  ~~  B ) }
44 ssdomg 7082 . . . . 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 1382 . . . 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 7095 . . . 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 6038 . . . . . . 7  |-  ( A  ^m  B )  e. 
_V
4948mptex 5898 . . . . . 6  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  e.  _V
5049rnex 5066 . . . . 5  |-  ran  (
f  e.  ( A  ^m  B )  |->  ran  f )  e.  _V
51 ensym 7085 . . . . . . . . . . . 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 7046 . . . . . . . . . . 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 5607 . . . . . . . . . . . . . . . 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 5532 . . . . . . . . . . . . . . 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 6960 . . . . . . . . . . . . . . . 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 5614 . . . . . . . . . . . . . . . 16  |-  ( f : B -1-1-onto-> x  ->  f : B -onto-> x )
65 forn 5589 . . . . . . . . . . . . . . . 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 2385 . . . . . . . . . . . . 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 1629 . . . . . . . . . 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 2648 . . . . . . . . 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 3352 . . . . . 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 2380 . . . . . . 7  |-  ( f  e.  ( A  ^m  B )  |->  ran  f
)  =  ( f  e.  ( A  ^m  B )  |->  ran  f
)
7877rnmpt 5049 . . . . . 6  |-  ran  (
f  e.  ( A  ^m  B )  |->  ran  f )  =  {
x  |  E. f  e.  ( A  ^m  B
) x  =  ran  f }
7976, 78syl6sseqr 3331 . . . . 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 7082 . . . . 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 2895 . . . . . . . . 9  |-  f  e. 
_V
8382rnex 5066 . . . . . . . 8  |-  ran  f  e.  _V
8483rgenw 2709 . . . . . . 7  |-  A. f  e.  ( A  ^m  B
) ran  f  e.  _V
8577fnmpt 5504 . . . . . . 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 5592 . . . . . 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 7864 . . . . 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 7089 . . . 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 7156 . . 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 4852 . . . . 5  |-  ( om  ~<_  A  ->  A  e.  _V )
96953ad2ant1 978 . . . 4  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  A  e.  _V )
97 map0e 6980 . . . 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 6811 . . . . . 6  |-  1o  e.  om
10099elexi 2901 . . . . 5  |-  1o  e.  _V
101100enref 7069 . . . 4  |-  1o  ~~  1o
102 df-sn 3756 . . . . 5  |-  { (/) }  =  { x  |  x  =  (/) }
103 df1o2 6665 . . . . 5  |-  1o  =  { (/) }
104 en0 7099 . . . . . . . 8  |-  ( x 
~~  (/)  <->  x  =  (/) )
105104anbi2i 676 . . . . . . 7  |-  ( ( x  C_  A  /\  x  ~~  (/) )  <->  ( x  C_  A  /\  x  =  (/) ) )
106 0ss 3592 . . . . . . . . 9  |-  (/)  C_  A
107 sseq1 3305 . . . . . . . . 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 2492 . . . . 5  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  { x  |  x  =  (/) }
112102, 103, 1113eqtr4ri 2411 . . . 4  |-  { x  |  ( x  C_  A  /\  x  ~~  (/) ) }  =  1o
113101, 112breqtrri 4171 . . 3  |-  1o  ~~  { x  |  ( x 
C_  A  /\  x  ~~  (/) ) }
11498, 113syl6eqbr 4183 . 2  |-  ( ( om  ~<_  A  /\  B  ~<_  A  /\  ( A  ^m  B )  e.  dom  card )  ->  ( A  ^m  (/) )  ~~  {
x  |  ( x 
C_  A  /\  x  ~~  (/) ) } )
1155, 94, 114pm2.61ne 2618 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 1547    = wceq 1649    e. wcel 1717   {cab 2366    =/= wne 2543   A.wral 2642   E.wrex 2643   _Vcvv 2892    C_ wss 3256   (/)c0 3564   {csn 3750   class class class wbr 4146    e. cmpt 4200   omcom 4778    X. cxp 4809   dom cdm 4811   ran crn 4812   Fun wfun 5381    Fn wfn 5382   -->wf 5383   -onto->wfo 5385   -1-1-onto->wf1o 5386  (class class class)co 6013   1oc1o 6646    ^m cmap 6947    ~~ cen 7035    ~<_ cdom 7036   cardccrd 7748
This theorem is referenced by:  infmap  8377
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-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-oi 7405  df-card 7752  df-acn 7755
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