Theorem rescval 17842

Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypothesis

Ref	Expression
rescval.1	⊢ 𝐷 = (𝐶 ↾cat 𝐻)

Assertion

Ref	Expression
rescval	⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Proof of Theorem rescval

Dummy variables ℎ 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rescval.1	. 2 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
2		elex 3484	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		elex 3484	. . 3 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
4		simpl 482	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
5		simpr 484	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻)
6	5	dmeqd 5896	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom ℎ = dom 𝐻)
7	6	dmeqd 5896	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom dom ℎ = dom dom 𝐻)
8	4, 7	oveq12d 7431	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑐 ↾s dom dom ℎ) = (𝐶 ↾s dom dom 𝐻))
9	5	opeq2d 4860	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ⟨(Hom ‘ndx), ℎ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
10	8, 9	oveq12d 7431	. . . 4 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((𝑐 ↾s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩) = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11		df-resc 17826	. . . 4 ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
12		ovex 7446	. . . 4 ⊢ ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
13	10, 11, 12	ovmpoa 7570	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
14	2, 3, 13	syl2an 596	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
15	1, 14	eqtrid 2781	1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3463  ⟨cop 4612  dom cdm 5665  ‘cfv 6541  (class class class)co 7413   sSet csts 17182  ndxcnx 17212   ↾_s cress 17252  Hom chom 17284   ↾_cat cresc 17823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-resc 17826

This theorem is referenced by:  rescval2  17843