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Theorem fvopab6 5621
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 2796 . . 3  |-  ( A  e.  D  ->  A  e.  _V )
2 fvopab6.2 . . . . 5  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
3 fvopab6.3 . . . . . 6  |-  ( x  =  A  ->  B  =  C )
43eqeq2d 2294 . . . . 5  |-  ( x  =  A  ->  (
y  =  B  <->  y  =  C ) )
52, 4anbi12d 691 . . . 4  |-  ( x  =  A  ->  (
( ph  /\  y  =  B )  <->  ( ps  /\  y  =  C ) ) )
6 iba 489 . . . . 5  |-  ( y  =  C  ->  ( ps 
<->  ( ps  /\  y  =  C ) ) )
76bicomd 192 . . . 4  |-  ( y  =  C  ->  (
( ps  /\  y  =  C )  <->  ps )
)
8 moeq 2941 . . . . . 6  |-  E* y 
y  =  B
98moani 2195 . . . . 5  |-  E* y
( ph  /\  y  =  B )
109a1i 10 . . . 4  |-  ( x  e.  _V  ->  E* y ( ph  /\  y  =  B )
)
11 fvopab6.1 . . . . 5  |-  F  =  { <. x ,  y
>.  |  ( ph  /\  y  =  B ) }
12 vex 2791 . . . . . . 7  |-  x  e. 
_V
1312biantrur 492 . . . . . 6  |-  ( (
ph  /\  y  =  B )  <->  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) )
1413opabbii 4083 . . . . 5  |-  { <. x ,  y >.  |  (
ph  /\  y  =  B ) }  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
1511, 14eqtri 2303 . . . 4  |-  F  =  { <. x ,  y
>.  |  ( x  e.  _V  /\  ( ph  /\  y  =  B ) ) }
165, 7, 10, 15fvopab3ig 5599 . . 3  |-  ( ( A  e.  _V  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
171, 16sylan 457 . 2  |-  ( ( A  e.  D  /\  C  e.  R )  ->  ( ps  ->  ( F `  A )  =  C ) )
18173impia 1148 1  |-  ( ( A  e.  D  /\  C  e.  R  /\  ps )  ->  ( F `
 A )  =  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E*wmo 2144   _Vcvv 2788   {copab 4076   ` cfv 5255
This theorem is referenced by:  cur1val  25198  istopx  25547
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 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263
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