Theorem rescval 18034

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 3475	. . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V)
3		elex 3475	. . 3 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V)
4		simpl 488	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶)
5		simpr 490	. . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻)
6	5	dmeqd 5895	. . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom ℎ = dom 𝐻)
7	6	dmeqd 5895	. . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom dom ℎ = dom dom 𝐻)
8	4, 7	oveq12d 7464	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑐 ↾s dom dom ℎ) = (𝐶 ↾s dom dom 𝐻))
9	5	opeq2d 4850	. . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ⟨(Hom ‘ndx), ℎ⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
10	8, 9	oveq12d 7464	. . . 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 18018	. . . 4 ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet ⟨(Hom ‘ndx), ℎ⟩))
12		ovex 7479	. . . 4 ⊢ ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) ∈ V
13	10, 11, 12	ovmpoa 7602	. . 3 ⊢ ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
14	2, 3, 13	syl2an 608	. 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
15	1, 14	eqtrid 2810	1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4838  dom cdm 5656  ‘cfv 6561  (class class class)co 7446   sSet csts 17373  ndxcnx 17403   ↾_s cress 17440  Hom chom 17471   ↾_cat cresc 18015

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-resc 18018

This theorem is used by:  rescval2  18035