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Theorem feqmptdf 23478
Description: Deduction form of dffn5f 5684. (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 5495 . . 3  |-  ( F : A --> B  ->  F  Fn  A )
31, 2syl 15 . 2  |-  ( ph  ->  F  Fn  A )
4 fnrel 5447 . . . . . 6  |-  ( F  Fn  A  ->  Rel  F )
5 feqmptdf.2 . . . . . . 7  |-  F/_ x F
6 nfcv 2502 . . . . . . 7  |-  F/_ y F
75, 6dfrel4 23441 . . . . . 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 5445 . . . . . 6  |-  F/ x  F  Fn  A
11 nfv 1624 . . . . . 6  |-  F/ y  F  Fn  A
12 fnbr 5451 . . . . . . . . 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 2368 . . . . . . . . 9  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
16 fnbrfvb 5670 . . . . . . . . 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 4183 . . . . 5  |-  ( F  Fn  A  ->  { <. x ,  y >.  |  x F y }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
218, 20eqtrd 2398 . . . 4  |-  ( F  Fn  A  ->  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) } )
22 df-mpt 4181 . . . 4  |-  ( x  e.  A  |->  ( F `
 x ) )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  ( F `  x ) ) }
2321, 22syl6eqr 2416 . . 3  |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
24 fvex 5646 . . . . . 6  |-  ( F `
 x )  e. 
_V
2524rgenw 2695 . . . . 5  |-  A. x  e.  A  ( F `  x )  e.  _V
269fnmptf 23477 . . . . 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 5438 . . . 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 1647    e. wcel 1715   F/_wnfc 2489   A.wral 2628   _Vcvv 2873   class class class wbr 4125   {copab 4178    e. cmpt 4179   Rel wrel 4797    Fn wfn 5353   -->wf 5354   ` cfv 5358
This theorem is referenced by:  esumf1o  23910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-fv 5366
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