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Theorem fvopab3g 5803
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 2497 . . . 4  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3g.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 693 . . 3  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3g.3 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 686 . . 3  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4474 . 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 5570 . . . . 5  |-  F  Fn  C
10 fnopfvb 5769 . . . . 5  |-  ( ( F  Fn  C  /\  A  e.  C )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
119, 10mpan 653 . . . 4  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  F ) )
128eleq2i 2501 . . . 4  |-  ( <. A ,  B >.  e.  F  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
1311, 12syl6bb 254 . . 3  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) )
1413adantr 453 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) } ) )
15 ibar 492 . . 3  |-  ( A  e.  C  ->  ( ch 
<->  ( A  e.  C  /\  ch ) ) )
1615adantr 453 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  <->  ( A  e.  C  /\  ch )
) )
176, 14, 163bitr4d 278 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726   E!weu 2282   <.cop 3818   {copab 4266    Fn wfn 5450   ` cfv 5455
This theorem is referenced by:  recmulnq  8842
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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