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Theorem fvopab3ig 5362
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 2150 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3ig.1 . . . . . . . 8  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 457 . . . . . . 7  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3ig.2 . . . . . . . 8  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 452 . . . . . . 7  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4086 . . . . . 6  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }  <->  ( A  e.  C  /\  ch )
) )
76biimpar 291 . . . . 5  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  ( A  e.  C  /\  ch )
)  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
87exp43 364 . . . 4  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( A  e.  C  -> 
( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) ) )
98pm2.43a 50 . . 3  |-  ( A  e.  C  ->  ( B  e.  D  ->  ( ch  ->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) ) )
109imp 122 . 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 5290 . . 3  |-  ( F `
 A )  =  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)
13 funopab 5035 . . . . 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 2023 . . . . . 6  |-  ( E* y ( x  e.  C  /\  ph )  <->  ( x  e.  C  ->  E* y ph ) )
1614, 15mpbir 144 . . . . 5  |-  E* y
( x  e.  C  /\  ph )
1713, 16mpgbir 1387 . . . 4  |-  Fun  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }
18 funopfv 5328 . . . 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 7 . . 3  |-  ( <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }  ->  ( { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } `  A
)  =  B )
2012, 19syl5eq 2132 . 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   E*wmo 1949   <.cop 3444   {copab 3890   Fun wfun 4996   ` cfv 5002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010
This theorem is referenced by:  fvmptg  5364  fvopab6  5380  ov6g  5764
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