Theorem dffn5f 6252

Description: Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)

Hypothesis

Ref	Expression
dffn5f.1	|- F/_ x F

Assertion

Ref	Expression
dffn5f	|- ( F Fn A <-> F = ( x e. A |-> ( F ` x ) ) )

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem dffn5f

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5 6241	. 2 |- ( F Fn A <-> F = ( z e. A |-> ( F ` z ) ) )
2		dffn5f.1	. . . . 5 |- F/_ x F
3		nfcv 2764	. . . . 5 |- F/_ x z
4	2, 3	nffv 6198	. . . 4 |- F/_ x ( F ` z )
5		nfcv 2764	. . . 4 |- F/_ z ( F ` x )
6		fveq2 6191	. . . 4 |- ( z = x -> ( F ` z ) = ( F ` x ) )
7	4, 5, 6	cbvmpt 4749	. . 3 |- ( z e. A |-> ( F ` z ) ) = ( x e. A |-> ( F ` x ) )
8	7	eqeq2i 2634	. 2 |- ( F = ( z e. A |-> ( F ` z ) ) <-> F = ( x e. A |-> ( F ` x ) ) )
9	1, 8	bitri 264	1 |- ( F Fn A <-> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   <-> wb 196   = wceq 1483   F/_wnfc 2751   |-> cmpt 4729   Fn wfn 5883   ` 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-fv 5896

This theorem is referenced by:  prdsgsum  18377  lgamgulm2  24762  fcomptf  29458  esumsup  30151  poimirlem16  33425  poimirlem19  33428  refsum2cnlem1  39196  etransclem2  40453