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Theorem fvopab6 5396
Description: Value of a function given by ordered-pair class abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
fvopab6.1  |-  F  =  { <. x ,  y
>.  |  ( ph  /\  y  =  B ) }
fvopab6.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
fvopab6.3  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
fvopab6  |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `
 A )  =  C )
Distinct variable groups:    x, A, y    ps, x, y    y, B   
x, C, y
Allowed substitution hints:    ph( x, y)    B( x)    D( x, y)    R( x, y)    F( x, y)

Proof of Theorem fvopab6
StepHypRef Expression
1 elex 2630 . . 3  |-  ( A  e.  D  ->  A  e.  _V )
2 fvopab6.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 fvopab6.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
43eqeq2d 2099 . . . . 5  |-  ( x  =  A  ->  (
y  =  B  <->  y  =  C ) )
52, 4anbi12d 457 . . . 4  |-  ( x  =  A  ->  (
( ph  /\  y  =  B )  <->  ( ps  /\  y  =  C ) ) )
6 iba 294 . . . . 5  |-  ( y  =  C  ->  ( ps 
<->  ( ps  /\  y  =  C ) ) )
76bicomd 139 . . . 4  |-  ( y  =  C  ->  (
( ps  /\  y  =  C )  <->  ps )
)
8 moeq 2790 . . . . . 6  |-  E* y 
y  =  B
98moani 2018 . . . . 5  |-  E* y
( ph  /\  y  =  B )
109a1i 9 . . . 4  |-  ( x  e.  _V  ->  E* y ( ph  /\  y  =  B )
)
11 fvopab6.1 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( ph  /\  y  =  B ) }
12 vex 2622 . . . . . . 7  |-  x  e. 
_V
1312biantrur 297 . . . . . 6  |-  ( (
ph  /\  y  =  B )  <->  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) )
1413opabbii 3905 . . . . 5  |-  { <. x ,  y >.  |  (
ph  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
1511, 14eqtri 2108 . . . 4  |-  F  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
165, 7, 10, 15fvopab3ig 5378 . . 3  |-  ( ( A  e.  _V  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
171, 16sylan 277 . 2  |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
18173impia 1140 1  |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `
 A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    /\ w3a 924    = wceq 1289    e. wcel 1438   E*wmo 1949   _Vcvv 2619   {copab 3898   ` cfv 5015
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 3957  ax-pow 4009  ax-pr 4036
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 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023
This theorem is referenced by: (None)
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