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Theorem fvopab6 5981
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 3118 . . 3  |-  ( A  e.  D  ->  A  e.  _V )
2 fvopab6.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 fvopab6.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
43eqeq2d 2471 . . . . 5  |-  ( x  =  A  ->  (
y  =  B  <->  y  =  C ) )
52, 4anbi12d 710 . . . 4  |-  ( x  =  A  ->  (
( ph  /\  y  =  B )  <->  ( ps  /\  y  =  C ) ) )
6 iba 503 . . . . 5  |-  ( y  =  C  ->  ( ps 
<->  ( ps  /\  y  =  C ) ) )
76bicomd 201 . . . 4  |-  ( y  =  C  ->  (
( ps  /\  y  =  C )  <->  ps )
)
8 moeq 3275 . . . . . 6  |-  E* y 
y  =  B
98moani 2346 . . . . 5  |-  E* y
( ph  /\  y  =  B )
109a1i 11 . . . 4  |-  ( x  e.  _V  ->  E* y ( ph  /\  y  =  B )
)
11 fvopab6.1 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( ph  /\  y  =  B ) }
12 vex 3112 . . . . . . 7  |-  x  e. 
_V
1312biantrur 506 . . . . . 6  |-  ( (
ph  /\  y  =  B )  <->  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) )
1413opabbii 4521 . . . . 5  |-  { <. x ,  y >.  |  (
ph  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
1511, 14eqtri 2486 . . . 4  |-  F  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
165, 7, 10, 15fvopab3ig 5953 . . 3  |-  ( ( A  e.  _V  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
171, 16sylan 471 . 2  |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
18173impia 1193 1  |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `
 A )  =  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   E*wmo 2284   _Vcvv 3109   {copab 4514   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602
This theorem is referenced by: (None)
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