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Theorem feqmptdf 25978
Description: Deduction form of dffn5f 5746. (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 5559 . 2  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrel 5509 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
4 feqmptdf.2 . . . . . 6  |-  F/_ x F
5 nfcv 2579 . . . . . 6  |-  F/_ y F
64, 5dfrel4 25937 . . . . 5  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
73, 6sylib 196 . . . 4  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
8 feqmptdf.1 . . . . . 6  |-  F/_ x A
94, 8nffn 5507 . . . . 5  |-  F/ x  F  Fn  A
10 nfv 1673 . . . . 5  |-  F/ y  F  Fn  A
11 fnbr 5513 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1211ex 434 . . . . . . 7  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
1312pm4.71rd 635 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
14 eqcom 2445 . . . . . . . 8  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
15 fnbrfvb 5732 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
1614, 15syl5bb 257 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
1716pm5.32da 641 . . . . . 6  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
1813, 17bitr4d 256 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
199, 10, 18opabbid 4354 . . . 4  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
207, 19eqtrd 2475 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
21 df-mpt 4352 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
2220, 21syl6eqr 2493 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
231, 2, 223syl 20 1  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   F/_wnfc 2566   class class class wbr 4292   {copab 4349    e. cmpt 4350   Rel wrel 4845    Fn wfn 5413   -->wf 5414   ` cfv 5418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426
This theorem is referenced by:  esumf1o  26504
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