Theorem feqresmptf 45831

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 2931	. . 3 ⊢ Ⅎ𝑥𝐶
2		feqresmptf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2, 1	nfres 5978	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶)
4		feqresmptf.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5		feqresmptf.3	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6	4, 5	fssresd 6743	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
7	1, 3, 6	feqmptdf 6949	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
8		fvres 6898	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
9	8	mpteq2ia 5207	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
10	7, 9	eqtrdi 2820	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1567  Ⅎwnfc 2916   ⊆ wss 3913   ↦ cmpt 5193   ↾ cres 5661  ⟶wf 6529  ‘cfv 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541

This theorem is referenced by: (None)