Theorem feqresmptf 44668

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 2892	. . 3 ⊢ Ⅎ𝑥𝐶
2		feqresmptf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
3	2, 1	nfres 5981	. . 3 ⊢ Ⅎ𝑥(𝐹 ↾ 𝐶)
4		feqresmptf.2	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
5		feqresmptf.3	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ 𝐴)
6	4, 5	fssresd 6759	. . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶𝐵)
7	1, 3, 6	feqmptdf 6964	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)))
8		fvres 6911	. . 3 ⊢ (𝑥 ∈ 𝐶 → ((𝐹 ↾ 𝐶)‘𝑥) = (𝐹‘𝑥))
9	8	mpteq2ia 5246	. 2 ⊢ (𝑥 ∈ 𝐶 ↦ ((𝐹 ↾ 𝐶)‘𝑥)) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥))
10	7, 9	eqtrdi 2781	1 ⊢ (𝜑 → (𝐹 ↾ 𝐶) = (𝑥 ∈ 𝐶 ↦ (𝐹‘𝑥)))

Syntax hints:   → wi 4   = wceq 1533  Ⅎwnfc 2875   ⊆ wss 3939   ↦ cmpt 5226   ↾ cres 5674  ⟶wf 6539  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551

This theorem is referenced by: (None)