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Theorem feqmptdf 24106
Description: Deduction form of dffn5f 5810. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypotheses
Ref Expression
feqmptdf.1  |-  F/_ x A
feqmptdf.2  |-  F/_ x F
feqmptdf.3  |-  ( ph  ->  F : A --> B )
Assertion
Ref Expression
feqmptdf  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )

Proof of Theorem feqmptdf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 feqmptdf.3 . 2  |-  ( ph  ->  F : A --> B )
2 ffn 5620 . 2  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrel 5572 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
4 feqmptdf.2 . . . . . 6  |-  F/_ x F
5 nfcv 2578 . . . . . 6  |-  F/_ y F
64, 5dfrel4 24065 . . . . 5  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
73, 6sylib 190 . . . 4  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
8 feqmptdf.1 . . . . . 6  |-  F/_ x A
94, 8nffn 5570 . . . . 5  |-  F/ x  F  Fn  A
10 nfv 1630 . . . . 5  |-  F/ y  F  Fn  A
11 fnbr 5576 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1211ex 425 . . . . . . 7  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
1312pm4.71rd 618 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
14 eqcom 2444 . . . . . . . 8  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
15 fnbrfvb 5796 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
1614, 15syl5bb 250 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
1716pm5.32da 624 . . . . . 6  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
1813, 17bitr4d 249 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
199, 10, 18opabbid 4295 . . . 4  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
207, 19eqtrd 2474 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
21 df-mpt 4293 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
2220, 21syl6eqr 2492 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
231, 2, 223syl 19 1  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   F/_wnfc 2565   class class class wbr 4237   {copab 4290    e. cmpt 4291   Rel wrel 4912    Fn wfn 5478   -->wf 5479   ` cfv 5483
This theorem is referenced by:  esumf1o  24476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491
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