Theorem rescval 16883

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 3414	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		elex 3414	. . 3 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
4		simpl 476	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
5		simpr 479	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻)
6	5	dmeqd 5573	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom ℎ = dom 𝐻)
7	6	dmeqd 5573	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom dom ℎ = dom dom 𝐻)
8	4, 7	oveq12d 6942	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑐 ↾s dom dom ℎ) = (𝐶 ↾s dom dom 𝐻))
9	5	opeq2d 4645	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ⟨(Hom ‘ndx), ℎ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
10	8, 9	oveq12d 6942	. . . 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 16867	. . . 4 ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
12		ovex 6956	. . . 4 ⊢ ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
13	10, 11, 12	ovmpt2a 7070	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
14	2, 3, 13	syl2an 589	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
15	1, 14	syl5eq 2826	1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ⟨cop 4404  dom cdm 5357  ‘cfv 6137  (class class class)co 6924  ndxcnx 16263   sSet csts 16264   ↾_s cress 16267  Hom chom 16360   ↾_cat cresc 16864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-iota 6101  df-fun 6139  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-resc 16867

This theorem is referenced by:  rescval2  16884