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Theorem feqmptdf 28247
Description: Deduction form of dffn5f 5932. (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 5742 . 2  |-  ( F : A --> B  ->  F  Fn  A )
3 fnrel 5688 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
4 feqmptdf.2 . . . . . 6  |-  F/_ x F
5 nfcv 2584 . . . . . 6  |-  F/_ y F
64, 5dfrel4 28197 . . . . 5  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
73, 6sylib 199 . . . 4  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
8 feqmptdf.1 . . . . . 6  |-  F/_ x A
94, 8nffn 5686 . . . . 5  |-  F/ x  F  Fn  A
10 nfv 1751 . . . . 5  |-  F/ y  F  Fn  A
11 fnbr 5692 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1211ex 435 . . . . . . 7  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
1312pm4.71rd 639 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
14 eqcom 2431 . . . . . . . 8  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
15 fnbrfvb 5917 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
1614, 15syl5bb 260 . . . . . . 7  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
1716pm5.32da 645 . . . . . 6  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
1813, 17bitr4d 259 . . . . 5  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
199, 10, 18opabbid 4483 . . . 4  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
207, 19eqtrd 2463 . . 3  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
21 df-mpt 4481 . . 3  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
2220, 21syl6eqr 2481 . 2  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
231, 2, 223syl 18 1  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   F/_wnfc 2570   class class class wbr 4420   {copab 4478    |-> cmpt 4479   Rel wrel 4854    Fn wfn 5592   -->wf 5593   ` cfv 5597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-fv 5605
This theorem is referenced by:  esumf1o  28866
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