Theorem feqresmptf 45255

Description: Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
feqresmptf.1	⊢ Ⅎ𝑥𝐹
feqresmptf.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
feqresmptf.3	⊢ (𝜑 → 𝐶 ⊆ 𝐴)

Assertion

Ref	Expression
feqresmptf	⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem feqresmptf

Step	Hyp	Ref	Expression
1		nfcv 2898	. . 3 ⊢ Ⅎ𝑥𝐶
2		feqresmptf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2, 1	nfres 5968	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶)
4		feqresmptf.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5		feqresmptf.3	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6	4, 5	fssresd 6745	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
7	1, 3, 6	feqmptdf 6949	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
8		fvres 6895	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
9	8	mpteq2ia 5216	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
10	7, 9	eqtrdi 2786	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1540  Ⅎwnfc 2883   ⊆ wss 3926   ↦ cmpt 5201   ↾ cres 5656  ⟶wf 6527  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539

This theorem is referenced by: (None)