Theorem dffn5imf 5758

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 5748	. 2 |- ( F Fn A -> F = ( z e. A |-> ( F ` z ) ) )
2		dffn5imf.1	. . . 4 |- F/_ x F
3		nfcv 2392	. . . 4 |- F/_ x z
4	2, 3	nffv 5705	. . 3 |- F/_ x ( F ` z )
5		nfcv 2392	. . 3 |- F/_ z ( F ` x )
6		fveq2 5695	. . 3 |- ( z = x -> ( F ` z ) = ( F ` x ) )
7	4, 5, 6	cbvmpt 4226	. 2 |- ( z e. A |-> ( F ` z ) ) = ( x e. A |-> ( F ` x ) )
8	1, 7	eqtrdi 2287	1 |- ( F Fn A -> F = ( x e. A |-> ( F ` x ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   F/_wnfc 2379   |-> cmpt 4192   Fn wfn 5372   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385

This theorem is used by: (None)