Theorem rescval2 16311

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

Hypotheses

Ref	Expression
rescval.1	⊢ 𝐷 = (𝐶 ↾cat 𝐻)
rescval2.1	⊢ (𝜑 → 𝐶 ∈ 𝑉)
rescval2.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
rescval2.3	⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))

Assertion

Ref	Expression
rescval2	⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Proof of Theorem rescval2

Step	Hyp	Ref	Expression
1		rescval2.1	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
2		rescval2.3	. . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆))
3		rescval2.2	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
4		xpexg 6858	. . . . 5 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ∈ 𝑊) → (𝑆 × 𝑆) ∈ V)
5	3, 3, 4	syl2anc 691	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
6		fnex 6386	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
7	2, 5, 6	syl2anc 691	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
8		rescval.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
9	8	rescval 16310	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
10	1, 7, 9	syl2anc 691	. 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
11		fndm 5904	. . . . . . 7 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
12	2, 11	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
13	12	dmeqd 5248	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
14		dmxpid 5266	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
15	13, 14	syl6eq 2660	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
16	15	oveq2d 6565	. . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆))
17	16	oveq1d 6564	. 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
18	10, 17	eqtrd 2644	1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131   × cxp 5036  dom cdm 5038   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693   ↾_s cress 15696  Hom chom 15779   ↾_cat cresc 16291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-resc 16294

This theorem is referenced by:  rescbas  16312  reschom  16313  rescco  16315  rescabs  16316  rescabs2  16317  dfrngc2  41764  dfringc2  41810  rngcresringcat  41822  rngcrescrhm  41877  rngcrescrhmALTV  41896