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Theorem fvopab3ig 5568
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 23-Oct-1999.)
Hypotheses
Ref Expression
fvopab3ig.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
fvopab3ig.2  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
fvopab3ig.3  |-  ( x  e.  C  ->  E* y ph )
fvopab3ig.4  |-  F  =  { <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }
Assertion
Ref Expression
fvopab3ig  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  ->  ( F `  A )  =  B ) )
Distinct variable groups:    x, y, A   
x, B, y    x, C, y    ch, x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)    D( x, y)    F( x, y)

Proof of Theorem fvopab3ig
StepHypRef Expression
1 eleq1 2233 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3ig.1 . . . . . . . 8  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 470 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3ig.2 . . . . . . . 8  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 461 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4251 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }  <->  ( A  e.  C  /\  ch )
) )
76biimpar 295 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( A  e.  C  /\  ch )
)  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
87exp43 370 . . . 4  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( A  e.  C  -> 
( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) ) )
98pm2.43a 51 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) )
109imp 123 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) )
11 fvopab3ig.4 . . . 4  |-  F  =  { <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }
1211fveq1i 5495 . . 3  |-  ( F `
 A )  =  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)
13 funopab 5231 . . . . 5  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) } 
<-> 
A. x E* y
( x  e.  C  /\  ph ) )
14 fvopab3ig.3 . . . . . 6  |-  ( x  e.  C  ->  E* y ph )
15 moanimv 2094 . . . . . 6  |-  ( E* y ( x  e.  C  /\  ph )  <->  ( x  e.  C  ->  E* y ph ) )
1614, 15mpbir 145 . . . . 5  |-  E* y
( x  e.  C  /\  ph )
1713, 16mpgbir 1446 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }
18 funopfv 5534 . . . 4  |-  ( Fun 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }  ->  ( { <. x ,  y
>.  |  ( x  e.  C  /\  ph ) } `  A )  =  B ) )
1917, 18ax-mp 5 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }  ->  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)  =  B )
2012, 19eqtrid 2215 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }  ->  ( F `  A )  =  B )
2110, 20syl6 33 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  ->  ( F `  A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348   E*wmo 2020    e. wcel 2141   <.cop 3584   {copab 4047   Fun wfun 5190   ` cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204
This theorem is referenced by:  fvmptg  5570  fvopab6  5590  ov6g  5988
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