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Theorem fvopab3g 5377
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 2150 . . . 4  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3g.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 457 . . 3  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3g.3 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 452 . . 3  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4095 . 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 5138 . . . . 5  |-  F  Fn  C
10 fnopfvb 5346 . . . . 5  |-  ( ( F  Fn  C  /\  A  e.  C )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
119, 10mpan 415 . . . 4  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  F ) )
128eleq2i 2154 . . . 4  |-  ( <. A ,  B >.  e.  F  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
1311, 12syl6bb 194 . . 3  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) )
1413adantr 270 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) } ) )
15 ibar 295 . . 3  |-  ( A  e.  C  ->  ( ch 
<->  ( A  e.  C  /\  ch ) ) )
1615adantr 270 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  <->  ( A  e.  C  /\  ch )
) )
176, 14, 163bitr4d 218 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E!weu 1948   <.cop 3449   {copab 3898    Fn wfn 5010   ` 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-fn 5018  df-fv 5023
This theorem is referenced by:  recmulnqg  6950
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