Theorem dffn5imf 5633

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

Hypothesis

Ref	Expression
dffn5imf.1	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
dffn5imf	⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 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 5623	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
2		dffn5imf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfcv 2347	. . . 4 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 5585	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2347	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥)
6		fveq2 5575	. . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
7	4, 5, 6	cbvmpt 4138	. 2 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
8	1, 7	eqtrdi 2253	1 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1372  Ⅎwnfc 2334   ↦ cmpt 4104   Fn wfn 5265  ‘cfv 5270

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fn 5273  df-fv 5278

This theorem is referenced by: (None)