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Theorem fvopab3g 5598
Description: Value of a function given by ordered-pair class abstraction. (Contributed by NM, 6-Mar-1996.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
fvopab3g.2  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
fvopab3g.3  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
fvopab3g.4  |-  ( x  e.  C  ->  E! y ph )
fvopab3g.5  |-  F  =  { <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }
Assertion
Ref Expression
fvopab3g  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  ch ) )
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 fvopab3g
StepHypRef Expression
1 eleq1 2343 . . . 4  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3g.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 691 . . 3  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3g.3 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 684 . . 3  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4283 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) }  <->  ( A  e.  C  /\  ch )
) )
7 fvopab3g.4 . . . . . 6  |-  ( x  e.  C  ->  E! y ph )
8 fvopab3g.5 . . . . . 6  |-  F  =  { <. x ,  y
>.  |  ( x  e.  C  /\  ph ) }
97, 8fnopab 5368 . . . . 5  |-  F  Fn  C
10 fnopfvb 5564 . . . . 5  |-  ( ( F  Fn  C  /\  A  e.  C )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
119, 10mpan 651 . . . 4  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  F ) )
128eleq2i 2347 . . . 4  |-  ( <. A ,  B >.  e.  F  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
1311, 12syl6bb 252 . . 3  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) )
1413adantr 451 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) } ) )
15 ibar 490 . . 3  |-  ( A  e.  C  ->  ( ch 
<->  ( A  e.  C  /\  ch ) ) )
1615adantr 451 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  <->  ( A  e.  C  /\  ch )
) )
176, 14, 163bitr4d 276 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E!weu 2143   <.cop 3643   {copab 4076    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  recmulnq  8588
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-fn 5258  df-fv 5263
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