Theorem dffn5imf 5570

Description: Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)

Hypothesis

Ref	Expression
dffn5imf.1	|- F/_ x F

Assertion

Ref	Expression
dffn5imf	|- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Distinct variable group:   x, A

Allowed substitution hint:   F( x)

Proof of Theorem dffn5imf

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dffn5im 5560	. 2 |- ( F Fn A -> F = ( z e. A |-> ( F ` z ) ) )
2		dffn5imf.1	. . . 4 |- F/_ x F
3		nfcv 2319	. . . 4 |- F/_ x z
4	2, 3	nffv 5524	. . 3 |- F/_ x ( F ` z )
5		nfcv 2319	. . 3 |- F/_ z ( F ` x )
6		fveq2 5514	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
7	4, 5, 6	cbvmpt 4097	. 2 |- ( z e. A |-> ( F ` z ) ) = ( x e. A |-> ( F ` x ) )
8	1, 7	eqtrdi 2226	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1353   F/_wnfc 2306   |-> cmpt 4063   Fn wfn 5210   ` cfv 5215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223

This theorem is referenced by: (None)