Theorem feqmptdf 6251

Description: Deduction form of dffn5f 6252. (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.

Step	Hyp	Ref	Expression
1		feqmptdf.3	. 2 |- ( ph -> F : A --> B )
2		ffn 6045	. 2 |- ( F : A --> B -> F Fn A )
3		fnrel 5989	. . . . 5 |- ( F Fn A -> Rel F )
4		feqmptdf.2	. . . . . 6 |- F/_ x F
5		nfcv 2764	. . . . . 6 |- F/_ y F
6	4, 5	dfrel4 5585	. . . . 5 |- ( Rel F <-> F = { <. x , y >. | x F y } )
7	3, 6	sylib 208	. . . 4 |- ( F Fn A -> F = { <. x , y >. | x F y } )
8		feqmptdf.1	. . . . . 6 |- F/_ x A
9	4, 8	nffn 5987	. . . . 5 |- F/ x F Fn A
10		nfv 1843	. . . . 5 |- F/ y F Fn A
11		fnbr 5993	. . . . . . . 8 |- ( ( F Fn A /\ x F y ) -> x e. A )
12	11	ex 450	. . . . . . 7 |- ( F Fn A -> ( x F y -> x e. A ) )
13	12	pm4.71rd 667	. . . . . 6 |- ( F Fn A -> ( x F y <-> ( x e. A /\ x F y ) ) )
14		eqcom 2629	. . . . . . . 8 |- ( y = ( F ` x ) <-> ( F ` x ) = y )
15		fnbrfvb 6236	. . . . . . . 8 |- ( ( F Fn A /\ x e. A ) -> ( ( F ` x ) = y <-> x F y ) )
16	14, 15	syl5bb 272	. . . . . . 7 |- ( ( F Fn A /\ x e. A ) -> ( y = ( F ` x ) <-> x F y ) )
17	16	pm5.32da 673	. . . . . 6 |- ( F Fn A -> ( ( x e. A /\ y = ( F ` x ) ) <-> ( x e. A /\ x F y ) ) )
18	13, 17	bitr4d 271	. . . . 5 |- ( F Fn A -> ( x F y <-> ( x e. A /\ y = ( F ` x ) ) ) )
19	9, 10, 18	opabbid 4715	. . . 4 |- ( F Fn A -> { <. x , y >. | x F y } = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
20	7, 19	eqtrd 2656	. . 3 |- ( F Fn A -> F = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) } )
21		df-mpt 4730	. . 3 |- ( x e. A |-> ( F ` x ) ) = { <. x , y >. | ( x e. A /\ y = ( F ` x ) ) }
22	20, 21	syl6eqr 2674	. 2 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )
23	1, 2, 22	3syl 18	1 |- ( ph -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   class class class wbr 4653   {copab 4712   |-> cmpt 4729   Rel wrel 5119   Fn wfn 5883   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  esumf1o  30112  feqresmptf  39433  liminfvaluz3  40028  liminfvaluz4  40031  volioofmpt  40211  volicofmpt  40214