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Theorem fvopab3ig 5928
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 2526 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3ig.1 . . . . . . . 8  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 708 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3ig.2 . . . . . . . 8  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 701 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4754 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }  <->  ( A  e.  C  /\  ch )
) )
76biimpar 483 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( A  e.  C  /\  ch )
)  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
87exp43 610 . . . 4  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( A  e.  C  -> 
( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) ) )
98pm2.43a 49 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) )
109imp 427 . 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 5849 . . 3  |-  ( F `
 A )  =  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)
13 funopab 5603 . . . . 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 2349 . . . . . 6  |-  ( E* y ( x  e.  C  /\  ph )  <->  ( x  e.  C  ->  E* y ph ) )
1614, 15mpbir 209 . . . . 5  |-  E* y
( x  e.  C  /\  ph )
1713, 16mpgbir 1627 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }
18 funopfv 5887 . . . 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, 19syl5eq 2507 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   E*wmo 2285   <.cop 4022   {copab 4496   Fun wfun 5564   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578
This theorem is referenced by:  fvmptg  5929  fvopab6  5956  ov6g  6413
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