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Theorem wemapso2OLD 7787
Description: An alternative to having a well-order on  R in wemapso 7786 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.) Obsolete version of wemapso2 7789 as of 1-Jul-2019. (New usage is discouraged.)
Hypotheses
Ref Expression
wemapso.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
wemapso2OLD.u  |-  U  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  { Z } ) )  e. 
Fin }
Assertion
Ref Expression
wemapso2OLD  |-  ( ( A  e.  V  /\  R  Or  A  /\  S  Or  B )  ->  T  Or  U )
Distinct variable groups:    x, B    x, w, y, z, A   
w, R, x, y, z    w, S, x, y, z    x, Z
Allowed substitution hints:    B( y, z, w)    T( x, y, z, w)    U( x, y, z, w)    V( x, y, z, w)    Z( y, z, w)

Proof of Theorem wemapso2OLD
Dummy variables  a 
b  c  d are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3002 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 wemapso.t . . 3  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
3 wemapso2OLD.u . . . 4  |-  U  =  { x  e.  ( B  ^m  A )  |  ( `' x " ( _V  \  { Z } ) )  e. 
Fin }
4 ssrab2 3458 . . . 4  |-  { x  e.  ( B  ^m  A
)  |  ( `' x " ( _V 
\  { Z }
) )  e.  Fin } 
C_  ( B  ^m  A )
53, 4eqsstri 3407 . . 3  |-  U  C_  ( B  ^m  A )
6 simp1 988 . . 3  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  ->  A  e.  _V )
7 simp2 989 . . 3  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  ->  R  Or  A )
8 simp3 990 . . 3  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  ->  S  Or  B )
9 simprll 761 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
a  e.  U )
10 cnveq 5034 . . . . . . . . . . 11  |-  ( x  =  a  ->  `' x  =  `' a
)
1110imaeq1d 5189 . . . . . . . . . 10  |-  ( x  =  a  ->  ( `' x " ( _V 
\  { Z }
) )  =  ( `' a " ( _V  \  { Z }
) ) )
1211eleq1d 2509 . . . . . . . . 9  |-  ( x  =  a  ->  (
( `' x "
( _V  \  { Z } ) )  e. 
Fin 
<->  ( `' a "
( _V  \  { Z } ) )  e. 
Fin ) )
1312, 3elrab2 3140 . . . . . . . 8  |-  ( a  e.  U  <->  ( a  e.  ( B  ^m  A
)  /\  ( `' a " ( _V  \  { Z } ) )  e.  Fin ) )
1413simprbi 464 . . . . . . 7  |-  ( a  e.  U  ->  ( `' a " ( _V  \  { Z }
) )  e.  Fin )
159, 14syl 16 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( `' a "
( _V  \  { Z } ) )  e. 
Fin )
16 simprlr 762 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
b  e.  U )
17 cnveq 5034 . . . . . . . . . . 11  |-  ( x  =  b  ->  `' x  =  `' b
)
1817imaeq1d 5189 . . . . . . . . . 10  |-  ( x  =  b  ->  ( `' x " ( _V 
\  { Z }
) )  =  ( `' b " ( _V  \  { Z }
) ) )
1918eleq1d 2509 . . . . . . . . 9  |-  ( x  =  b  ->  (
( `' x "
( _V  \  { Z } ) )  e. 
Fin 
<->  ( `' b "
( _V  \  { Z } ) )  e. 
Fin ) )
2019, 3elrab2 3140 . . . . . . . 8  |-  ( b  e.  U  <->  ( b  e.  ( B  ^m  A
)  /\  ( `' b " ( _V  \  { Z } ) )  e.  Fin ) )
2120simprbi 464 . . . . . . 7  |-  ( b  e.  U  ->  ( `' b " ( _V  \  { Z }
) )  e.  Fin )
2216, 21syl 16 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( `' b "
( _V  \  { Z } ) )  e. 
Fin )
23 unfi 7600 . . . . . 6  |-  ( ( ( `' a "
( _V  \  { Z } ) )  e. 
Fin  /\  ( `' b " ( _V  \  { Z } ) )  e.  Fin )  -> 
( ( `' a
" ( _V  \  { Z } ) )  u.  ( `' b
" ( _V  \  { Z } ) ) )  e.  Fin )
2415, 22, 23syl2anc 661 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( ( `' a
" ( _V  \  { Z } ) )  u.  ( `' b
" ( _V  \  { Z } ) ) )  e.  Fin )
255, 9sseldi 3375 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
a  e.  ( B  ^m  A ) )
26 elmapi 7255 . . . . . . . . 9  |-  ( a  e.  ( B  ^m  A )  ->  a : A --> B )
2725, 26syl 16 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
a : A --> B )
28 ffn 5580 . . . . . . . 8  |-  ( a : A --> B  -> 
a  Fn  A )
2927, 28syl 16 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
a  Fn  A )
305, 16sseldi 3375 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
b  e.  ( B  ^m  A ) )
31 elmapi 7255 . . . . . . . . 9  |-  ( b  e.  ( B  ^m  A )  ->  b : A --> B )
3230, 31syl 16 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
b : A --> B )
33 ffn 5580 . . . . . . . 8  |-  ( b : A --> B  -> 
b  Fn  A )
3432, 33syl 16 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
b  Fn  A )
35 fndmdif 5828 . . . . . . 7  |-  ( ( a  Fn  A  /\  b  Fn  A )  ->  dom  ( a  \ 
b )  =  {
c  e.  A  | 
( a `  c
)  =/=  ( b `
 c ) } )
3629, 34, 35syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  dom  ( a  \  b
)  =  { c  e.  A  |  ( a `  c )  =/=  ( b `  c ) } )
37 eqtr3 2462 . . . . . . . . . . 11  |-  ( ( ( a `  c
)  =  Z  /\  ( b `  c
)  =  Z )  ->  ( a `  c )  =  ( b `  c ) )
3837necon3ai 2675 . . . . . . . . . 10  |-  ( ( a `  c )  =/=  ( b `  c )  ->  -.  ( ( a `  c )  =  Z  /\  ( b `  c )  =  Z ) )
39 neorian 2720 . . . . . . . . . 10  |-  ( ( ( a `  c
)  =/=  Z  \/  ( b `  c
)  =/=  Z )  <->  -.  ( ( a `  c )  =  Z  /\  ( b `  c )  =  Z ) )
4038, 39sylibr 212 . . . . . . . . 9  |-  ( ( a `  c )  =/=  ( b `  c )  ->  (
( a `  c
)  =/=  Z  \/  ( b `  c
)  =/=  Z ) )
41 elun 3518 . . . . . . . . . 10  |-  ( c  e.  ( ( `' a " ( _V 
\  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) )  <->  ( c  e.  ( `' a "
( _V  \  { Z } ) )  \/  c  e.  ( `' b " ( _V 
\  { Z }
) ) ) )
4229adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  a  Fn  A )
43 elpreima 5844 . . . . . . . . . . . . . 14  |-  ( a  Fn  A  ->  (
c  e.  ( `' a " ( _V 
\  { Z }
) )  <->  ( c  e.  A  /\  (
a `  c )  e.  ( _V  \  { Z } ) ) ) )
4442, 43syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
c  e.  ( `' a " ( _V 
\  { Z }
) )  <->  ( c  e.  A  /\  (
a `  c )  e.  ( _V  \  { Z } ) ) ) )
45 simpr 461 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  c  e.  A )
4645biantrurd 508 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
( a `  c
)  e.  ( _V 
\  { Z }
)  <->  ( c  e.  A  /\  ( a `
 c )  e.  ( _V  \  { Z } ) ) ) )
4744, 46bitr4d 256 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
c  e.  ( `' a " ( _V 
\  { Z }
) )  <->  ( a `  c )  e.  ( _V  \  { Z } ) ) )
48 fvex 5722 . . . . . . . . . . . . 13  |-  ( a `
 c )  e. 
_V
49 eldifsn 4021 . . . . . . . . . . . . 13  |-  ( ( a `  c )  e.  ( _V  \  { Z } )  <->  ( (
a `  c )  e.  _V  /\  ( a `
 c )  =/= 
Z ) )
5048, 49mpbiran 909 . . . . . . . . . . . 12  |-  ( ( a `  c )  e.  ( _V  \  { Z } )  <->  ( a `  c )  =/=  Z
)
5147, 50syl6bb 261 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
c  e.  ( `' a " ( _V 
\  { Z }
) )  <->  ( a `  c )  =/=  Z
) )
5234adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  b  Fn  A )
53 elpreima 5844 . . . . . . . . . . . . . 14  |-  ( b  Fn  A  ->  (
c  e.  ( `' b " ( _V 
\  { Z }
) )  <->  ( c  e.  A  /\  (
b `  c )  e.  ( _V  \  { Z } ) ) ) )
5452, 53syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
c  e.  ( `' b " ( _V 
\  { Z }
) )  <->  ( c  e.  A  /\  (
b `  c )  e.  ( _V  \  { Z } ) ) ) )
5545biantrurd 508 . . . . . . . . . . . . 13  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
( b `  c
)  e.  ( _V 
\  { Z }
)  <->  ( c  e.  A  /\  ( b `
 c )  e.  ( _V  \  { Z } ) ) ) )
5654, 55bitr4d 256 . . . . . . . . . . . 12  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
c  e.  ( `' b " ( _V 
\  { Z }
) )  <->  ( b `  c )  e.  ( _V  \  { Z } ) ) )
57 fvex 5722 . . . . . . . . . . . . 13  |-  ( b `
 c )  e. 
_V
58 eldifsn 4021 . . . . . . . . . . . . 13  |-  ( ( b `  c )  e.  ( _V  \  { Z } )  <->  ( (
b `  c )  e.  _V  /\  ( b `
 c )  =/= 
Z ) )
5957, 58mpbiran 909 . . . . . . . . . . . 12  |-  ( ( b `  c )  e.  ( _V  \  { Z } )  <->  ( b `  c )  =/=  Z
)
6056, 59syl6bb 261 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
c  e.  ( `' b " ( _V 
\  { Z }
) )  <->  ( b `  c )  =/=  Z
) )
6151, 60orbi12d 709 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
( c  e.  ( `' a " ( _V  \  { Z }
) )  \/  c  e.  ( `' b "
( _V  \  { Z } ) ) )  <-> 
( ( a `  c )  =/=  Z  \/  ( b `  c
)  =/=  Z ) ) )
6241, 61syl5bb 257 . . . . . . . . 9  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
c  e.  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) )  <->  ( (
a `  c )  =/=  Z  \/  ( b `
 c )  =/= 
Z ) ) )
6340, 62syl5ibr 221 . . . . . . . 8  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Or  B
)  /\  ( (
a  e.  U  /\  b  e.  U )  /\  a  =/=  b
) )  /\  c  e.  A )  ->  (
( a `  c
)  =/=  ( b `
 c )  -> 
c  e.  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) ) )
6463ralrimiva 2820 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  A. c  e.  A  ( ( a `  c )  =/=  (
b `  c )  ->  c  e.  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) ) )
65 rabss 3450 . . . . . . 7  |-  ( { c  e.  A  | 
( a `  c
)  =/=  ( b `
 c ) } 
C_  ( ( `' a " ( _V 
\  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) )  <->  A. c  e.  A  ( (
a `  c )  =/=  ( b `  c
)  ->  c  e.  ( ( `' a
" ( _V  \  { Z } ) )  u.  ( `' b
" ( _V  \  { Z } ) ) ) ) )
6664, 65sylibr 212 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  { c  e.  A  |  ( a `  c )  =/=  (
b `  c ) }  C_  ( ( `' a " ( _V 
\  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )
6736, 66eqsstrd 3411 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  dom  ( a  \  b
)  C_  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )
68 ssfi 7554 . . . . 5  |-  ( ( ( ( `' a
" ( _V  \  { Z } ) )  u.  ( `' b
" ( _V  \  { Z } ) ) )  e.  Fin  /\  dom  ( a  \  b
)  C_  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )  ->  dom  ( a  \  b )  e. 
Fin )
6924, 67, 68syl2anc 661 . . . 4  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  dom  ( a  \  b
)  e.  Fin )
70 cnvimass 5210 . . . . . . . . 9  |-  ( `' a " ( _V 
\  { Z }
) )  C_  dom  a
71 fdm 5584 . . . . . . . . . 10  |-  ( a : A --> B  ->  dom  a  =  A
)
7227, 71syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  dom  a  =  A
)
7370, 72syl5sseq 3425 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( `' a "
( _V  \  { Z } ) )  C_  A )
74 cnvimass 5210 . . . . . . . . 9  |-  ( `' b " ( _V 
\  { Z }
) )  C_  dom  b
75 fdm 5584 . . . . . . . . . 10  |-  ( b : A --> B  ->  dom  b  =  A
)
7632, 75syl 16 . . . . . . . . 9  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  dom  b  =  A
)
7774, 76syl5sseq 3425 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( `' b "
( _V  \  { Z } ) )  C_  A )
7873, 77unssd 3553 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( ( `' a
" ( _V  \  { Z } ) )  u.  ( `' b
" ( _V  \  { Z } ) ) )  C_  A )
797adantr 465 . . . . . . 7  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  R  Or  A )
80 soss 4680 . . . . . . 7  |-  ( ( ( `' a "
( _V  \  { Z } ) )  u.  ( `' b "
( _V  \  { Z } ) ) ) 
C_  A  ->  ( R  Or  A  ->  R  Or  ( ( `' a " ( _V 
\  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) ) )
8178, 79, 80sylc 60 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  R  Or  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )
82 wofi 7582 . . . . . 6  |-  ( ( R  Or  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) )  /\  ( ( `' a
" ( _V  \  { Z } ) )  u.  ( `' b
" ( _V  \  { Z } ) ) )  e.  Fin )  ->  R  We  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )
8381, 24, 82syl2anc 661 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  R  We  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )
84 wefr 4731 . . . . 5  |-  ( R  We  ( ( `' a " ( _V 
\  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) )  ->  R  Fr  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )
8583, 84syl 16 . . . 4  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  R  Fr  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) ) )
86 simprr 756 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
a  =/=  b )
87 fndmdifeq0 5830 . . . . . . 7  |-  ( ( a  Fn  A  /\  b  Fn  A )  ->  ( dom  ( a 
\  b )  =  (/) 
<->  a  =  b ) )
8829, 34, 87syl2anc 661 . . . . . 6  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( dom  ( a  \  b )  =  (/) 
<->  a  =  b ) )
8988necon3bid 2637 . . . . 5  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  -> 
( dom  ( a  \  b )  =/=  (/) 
<->  a  =/=  b ) )
9086, 89mpbird 232 . . . 4  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  dom  ( a  \  b
)  =/=  (/) )
91 fri 4703 . . . 4  |-  ( ( ( dom  ( a 
\  b )  e. 
Fin  /\  R  Fr  ( ( `' a
" ( _V  \  { Z } ) )  u.  ( `' b
" ( _V  \  { Z } ) ) ) )  /\  ( dom  ( a  \  b
)  C_  ( ( `' a " ( _V  \  { Z }
) )  u.  ( `' b " ( _V  \  { Z }
) ) )  /\  dom  ( a  \  b
)  =/=  (/) ) )  ->  E. c  e.  dom  ( a  \  b
) A. d  e. 
dom  ( a  \ 
b )  -.  d R c )
9269, 85, 67, 90, 91syl22anc 1219 . . 3  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  /\  ( ( a  e.  U  /\  b  e.  U )  /\  a  =/=  b ) )  ->  E. c  e.  dom  ( a  \  b
) A. d  e. 
dom  ( a  \ 
b )  -.  d R c )
932, 5, 6, 7, 8, 92wemapsolem 7785 . 2  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  ->  T  Or  U )
941, 93syl3an1 1251 1  |-  ( ( A  e.  V  /\  R  Or  A  /\  S  Or  B )  ->  T  Or  U )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2620   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 2993    \ cdif 3346    u. cun 3347    C_ wss 3349   (/)c0 3658   {csn 3898   class class class wbr 4313   {copab 4370    Or wor 4661    Fr wfr 4697    We wwe 4699   `'ccnv 4860   dom cdm 4861   "cima 4864    Fn wfn 5434   -->wf 5435   ` cfv 5439  (class class class)co 6112    ^m cmap 7235   Fincfn 7331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-om 6498  df-1st 6598  df-2nd 6599  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-fin 7335
This theorem is referenced by: (None)
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