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Theorem wemapso2OLD 7989
Description: An alternative to having a well-order on  R in wemapso 7988 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.) Obsolete version of wemapso2 7991 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 3127 . 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 3590 . . . 4  |-  { x  e.  ( B  ^m  A
)  |  ( `' x " ( _V 
\  { Z }
) )  e.  Fin } 
C_  ( B  ^m  A )
53, 4eqsstri 3539 . . 3  |-  U  C_  ( B  ^m  A )
6 simp1 996 . . 3  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  ->  A  e.  _V )
7 simp2 997 . . 3  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  ->  R  Or  A )
8 simp3 998 . . 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 5182 . . . . . . . . . . 11  |-  ( x  =  a  ->  `' x  =  `' a
)
1110imaeq1d 5342 . . . . . . . . . 10  |-  ( x  =  a  ->  ( `' x " ( _V 
\  { Z }
) )  =  ( `' a " ( _V  \  { Z }
) ) )
1211eleq1d 2536 . . . . . . . . 9  |-  ( x  =  a  ->  (
( `' x "
( _V  \  { Z } ) )  e. 
Fin 
<->  ( `' a "
( _V  \  { Z } ) )  e. 
Fin ) )
1312, 3elrab2 3268 . . . . . . . 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 5182 . . . . . . . . . . 11  |-  ( x  =  b  ->  `' x  =  `' b
)
1817imaeq1d 5342 . . . . . . . . . 10  |-  ( x  =  b  ->  ( `' x " ( _V 
\  { Z }
) )  =  ( `' b " ( _V  \  { Z }
) ) )
1918eleq1d 2536 . . . . . . . . 9  |-  ( x  =  b  ->  (
( `' x "
( _V  \  { Z } ) )  e. 
Fin 
<->  ( `' b "
( _V  \  { Z } ) )  e. 
Fin ) )
2019, 3elrab2 3268 . . . . . . . 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 7799 . . . . . 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 3507 . . . . . . . . 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 7452 . . . . . . . . 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 5737 . . . . . . . 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 3507 . . . . . . . . 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 7452 . . . . . . . . 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 5737 . . . . . . . 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 5992 . . . . . . 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 2495 . . . . . . . . . . 11  |-  ( ( ( a `  c
)  =  Z  /\  ( b `  c
)  =  Z )  ->  ( a `  c )  =  ( b `  c ) )
3837necon3ai 2695 . . . . . . . . . 10  |-  ( ( a `  c )  =/=  ( b `  c )  ->  -.  ( ( a `  c )  =  Z  /\  ( b `  c )  =  Z ) )
39 neorian 2794 . . . . . . . . . 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 3650 . . . . . . . . . 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 6008 . . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . 13  |-  ( a `
 c )  e. 
_V
49 eldifsn 4158 . . . . . . . . . . . . 13  |-  ( ( a `  c )  e.  ( _V  \  { Z } )  <->  ( (
a `  c )  e.  _V  /\  ( a `
 c )  =/= 
Z ) )
5048, 49mpbiran 916 . . . . . . . . . . . 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 6008 . . . . . . . . . . . . . 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 5882 . . . . . . . . . . . . 13  |-  ( b `
 c )  e. 
_V
58 eldifsn 4158 . . . . . . . . . . . . 13  |-  ( ( b `  c )  e.  ( _V  \  { Z } )  <->  ( (
b `  c )  e.  _V  /\  ( b `
 c )  =/= 
Z ) )
5957, 58mpbiran 916 . . . . . . . . . . . 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 2881 . . . . . . 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 3582 . . . . . . 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 3543 . . . . 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 7752 . . . . 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 5363 . . . . . . . . 9  |-  ( `' a " ( _V 
\  { Z }
) )  C_  dom  a
71 fdm 5741 . . . . . . . . . 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 3557 . . . . . . . 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 5363 . . . . . . . . 9  |-  ( `' b " ( _V 
\  { Z }
) )  C_  dom  b
75 fdm 5741 . . . . . . . . . 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 3557 . . . . . . . 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 3685 . . . . . . 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 4824 . . . . . . 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 7781 . . . . . 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 4875 . . . . 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 5994 . . . . . . 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 2725 . . . . 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 4847 . . . 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 1229 . . 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 7987 . 2  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Or  B )  ->  T  Or  U )
941, 93syl3an1 1261 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118    \ cdif 3478    u. cun 3479    C_ wss 3481   (/)c0 3790   {csn 4033   class class class wbr 4453   {copab 4510    Or wor 4805    Fr wfr 4841    We wwe 4843   `'ccnv 5004   dom cdm 5005   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    ^m cmap 7432   Fincfn 7528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-en 7529  df-fin 7532
This theorem is referenced by: (None)
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