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Theorem fvopab3g 5460
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 2178 . . . 4  |-  ( x  =  A  ->  (
x  e.  C  <->  A  e.  C ) )
2 fvopab3g.2 . . . 4  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
31, 2anbi12d 462 . . 3  |-  ( x  =  A  ->  (
( x  e.  C  /\  ph )  <->  ( A  e.  C  /\  ps )
) )
4 fvopab3g.3 . . . 4  |-  ( y  =  B  ->  ( ps 
<->  ch ) )
54anbi2d 457 . . 3  |-  ( y  =  B  ->  (
( A  e.  C  /\  ps )  <->  ( A  e.  C  /\  ch )
) )
63, 5opelopabg 4158 . 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 5215 . . . . 5  |-  F  Fn  C
10 fnopfvb 5429 . . . . 5  |-  ( ( F  Fn  C  /\  A  e.  C )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e.  F ) )
119, 10mpan 418 . . . 4  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  F ) )
128eleq2i 2182 . . . 4  |-  ( <. A ,  B >.  e.  F  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } )
1311, 12syl6bb 195 . . 3  |-  ( A  e.  C  ->  (
( F `  A
)  =  B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ( x  e.  C  /\  ph ) } ) )
1413adantr 272 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  <. A ,  B >.  e. 
{ <. x ,  y
>.  |  ( x  e.  C  /\  ph ) } ) )
15 ibar 297 . . 3  |-  ( A  e.  C  ->  ( ch 
<->  ( A  e.  C  /\  ch ) ) )
1615adantr 272 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ch  <->  ( A  e.  C  /\  ch )
) )
176, 14, 163bitr4d 219 1  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( ( F `  A )  =  B  <->  ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   E!weu 1975   <.cop 3498   {copab 3956    Fn wfn 5086   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  recmulnqg  7163
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