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Theorem feqmptdf 23230
Description: Deduction form of dffn5f 5579. (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 . . 3  |-  ( ph  ->  F : A --> B )
2 ffn 5391 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 15 . 2  |-  ( ph  ->  F  Fn  A )
4 fnrel 5344 . . . . . 6  |-  ( F  Fn  A  ->  Rel  F )
5 feqmptdf.2 . . . . . . 7  |-  F/_ x F
6 nfcv 2421 . . . . . . 7  |-  F/_ y F
75, 6dfrel4 23206 . . . . . 6  |-  ( Rel 
F  <->  F  =  { <. x ,  y >.  |  x F y } )
84, 7sylib 188 . . . . 5  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  x F
y } )
9 feqmptdf.1 . . . . . . 7  |-  F/_ x A
105, 9nffn 5342 . . . . . 6  |-  F/ x  F  Fn  A
11 nfv 1607 . . . . . 6  |-  F/ y  F  Fn  A
12 fnbr 5348 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x F y )  ->  x  e.  A )
1312ex 423 . . . . . . . 8  |-  ( F  Fn  A  ->  (
x F y  ->  x  e.  A )
)
1413pm4.71rd 616 . . . . . . 7  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  x F y ) ) )
15 eqcom 2287 . . . . . . . . 9  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
16 fnbrfvb 5565 . . . . . . . . 9  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( ( F `  x )  =  y  <-> 
x F y ) )
1715, 16syl5bb 248 . . . . . . . 8  |-  ( ( F  Fn  A  /\  x  e.  A )  ->  ( y  =  ( F `  x )  <-> 
x F y ) )
1817pm5.32da 622 . . . . . . 7  |-  ( F  Fn  A  ->  (
( x  e.  A  /\  y  =  ( F `  x )
)  <->  ( x  e.  A  /\  x F y ) ) )
1914, 18bitr4d 247 . . . . . 6  |-  ( F  Fn  A  ->  (
x F y  <->  ( x  e.  A  /\  y  =  ( F `  x ) ) ) )
2010, 11, 19opabbid 4083 . . . . 5  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
218, 20eqtrd 2317 . . . 4  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
22 df-mpt 4081 . . . 4  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
2321, 22syl6eqr 2335 . . 3  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
24 fvex 5541 . . . . . 6  |-  ( F `
 x )  e. 
_V
2524rgenw 2612 . . . . 5  |-  A. x  e.  A  ( F `  x )  e.  _V
269fnmptf 23229 . . . . 5  |-  ( A. x  e.  A  ( F `  x )  e.  _V  ->  ( x  e.  A  |->  ( F `
 x ) )  Fn  A )
2725, 26ax-mp 8 . . . 4  |-  ( x  e.  A  |->  ( F `
 x ) )  Fn  A
28 fneq1 5335 . . . 4  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  -> 
( F  Fn  A  <->  ( x  e.  A  |->  ( F `  x ) )  Fn  A ) )
2927, 28mpbiri 224 . . 3  |-  ( F  =  ( x  e.  A  |->  ( F `  x ) )  ->  F  Fn  A )
3023, 29impbii 180 . 2  |-  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F `  x
) ) )
313, 30sylib 188 1  |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `
 x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686   F/_wnfc 2408   A.wral 2545   _Vcvv 2790   class class class wbr 4025   {copab 4078    e. cmpt 4079   Rel wrel 4696    Fn wfn 5252   -->wf 5253   ` cfv 5257
This theorem is referenced by:  esumf1o  23431
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4216
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-fv 5265
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