Theorem dffn5imf 5701

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 5691	. 2 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)))
2		dffn5imf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3		nfcv 2374	. . . 4 ⊢ Ⅎ𝑥𝑧
4	2, 3	nffv 5649	. . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧)
5		nfcv 2374	. . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥)
6		fveq2 5639	. . 3 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
7	4, 5, 6	cbvmpt 4184	. 2 ⊢ (𝑧 ∈ 𝐴 ↦ (𝐹‘𝑧)) = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))
8	1, 7	eqtrdi 2280	1 ⊢ (𝐹 Fn 𝐴 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1397  Ⅎwnfc 2361   ↦ cmpt 4150   Fn wfn 5321  ‘cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by: (None)