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Theorem fvopab3ig 5744
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 2449 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3ig.1 . . . . . . . 8  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 692 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3ig.2 . . . . . . . 8  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 685 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4416 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }  <->  ( A  e.  C  /\  ch )
) )
76biimpar 472 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( A  e.  C  /\  ch )
)  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
87exp43 596 . . . 4  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( A  e.  C  -> 
( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) ) )
98pm2.43a 47 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) )
109imp 419 . 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 5671 . . 3  |-  ( F `
 A )  =  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)
13 funopab 5428 . . . . 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 2298 . . . . . 6  |-  ( E* y ( x  e.  C  /\  ph )  <->  ( x  e.  C  ->  E* y ph ) )
1614, 15mpbir 201 . . . . 5  |-  E* y
( x  e.  C  /\  ph )
1713, 16mpgbir 1556 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }
18 funopfv 5707 . . . 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 8 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }  ->  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)  =  B )
2012, 19syl5eq 2433 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }  ->  ( F `  A )  =  B )
2110, 20syl6 31 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  ->  ( F `  A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E*wmo 2241   <.cop 3762   {copab 4208   Fun wfun 5390   ` cfv 5396
This theorem is referenced by:  fvmptg  5745  fvopab6  5767  ov6g  6152
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-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fv 5404
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