Theorem dffn5imf 5228

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 5219	. 2 |- ( F Fn A -> F = ( z e. A |-> ( F ` z ) ) )
2		dffn5imf.1	. . . 4 |- F/_ x F
3		nfcv 2178	. . . 4 |- F/_ x z
4	2, 3	nffv 5185	. . 3 |- F/_ x ( F ` z )
5		nfcv 2178	. . 3 |- F/_ z ( F ` x )
6		fveq2 5178	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
7	4, 5, 6	cbvmpt 3851	. 2 |- ( z e. A |-> ( F ` z ) ) = ( x e. A |-> ( F ` x ) )
8	1, 7	syl6eq 2088	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

Syntax hints:   -> wi 4   = wceq 1243   F/_wnfc 2165   |-> cmpt 3818   Fn wfn 4897   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)