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Theorem fvopab3g 5271
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 2142 . . . 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 4025 . 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 5048 . . . . 5  |-  F  Fn  C
10 fnopfvb 5241 . . . . 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 2146 . . . 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 1285    e. wcel 1434   E!weu 1942   <.cop 3403   {copab 3840    Fn wfn 4921   ` cfv 4926
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fn 4929  df-fv 4934
This theorem is referenced by:  recmulnqg  6632
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