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Theorem feqmptdf 27324
Description: Deduction form of dffn5f 5929. (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 5737 . 2  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrel 5685 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
4 feqmptdf.2 . . . . . 6  |-  F/_ x F
5 nfcv 2629 . . . . . 6  |-  F/_ y F
64, 5dfrel4 27277 . . . . 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 5683 . . . . 5  |-  F/ x  F  Fn  A
10 nfv 1683 . . . . 5  |-  F/ y  F  Fn  A
11 fnbr 5689 . . . . . . . 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 2476 . . . . . . . 8  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
15 fnbrfvb 5914 . . . . . . . 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 4515 . . . 4  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
207, 19eqtrd 2508 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
21 df-mpt 4513 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
2220, 21syl6eqr 2526 . 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 1379    e. wcel 1767   F/_wnfc 2615   class class class wbr 4453   {copab 4510    |-> cmpt 4511   Rel wrel 5010    Fn wfn 5589   -->wf 5590   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  esumf1o  27886
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