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Theorem fvopab3ig 5561
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 2344 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3ig.1 . . . . . . . 8  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 691 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3ig.2 . . . . . . . 8  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 684 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4282 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }  <->  ( A  e.  C  /\  ch )
) )
76biimpar 471 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( A  e.  C  /\  ch )
)  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
87exp43 595 . . . 4  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( A  e.  C  -> 
( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) ) )
98pm2.43a 45 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) )
109imp 418 . 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 5487 . . 3  |-  ( F `
 A )  =  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)
13 funopab 5253 . . . . 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 2202 . . . . . 6  |-  ( E* y ( x  e.  C  /\  ph )  <->  ( x  e.  C  ->  E* y ph ) )
1614, 15mpbir 200 . . . . 5  |-  E* y
( x  e.  C  /\  ph )
1713, 16mpgbir 1537 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }
18 funopfv 5524 . . . 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 2328 . 2  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }  ->  ( F `  A )  =  B )
2110, 20syl6 29 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  ->  ( F `  A )  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1685   E*wmo 2145   <.cop 3644   {copab 4077   Fun wfun 5215   ` cfv 5221
This theorem is referenced by:  fvmptg  5562  fvopab6  5583  ov6g  5947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fv 5229
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