Theorem rescval2 17781

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	3, 3	xpexd 7734	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
5		fnex 7213	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
6	2, 4, 5	syl2anc 583	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
7		rescval.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
8	7	rescval 17780	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
9	1, 6, 8	syl2anc 583	. 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
10	2	fndmd 6647	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
11	10	dmeqd 5898	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12		dmxpid 5922	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
13	11, 12	eqtrdi 2782	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
14	13	oveq2d 7420	. . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆))
15	14	oveq1d 7419	. 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
16	9, 15	eqtrd 2766	1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ⟨cop 4629   × cxp 5667  dom cdm 5669   Fn wfn 6531  ‘cfv 6536  (class class class)co 7404   sSet csts 17102  ndxcnx 17132   ↾_s cress 17179  Hom chom 17214   ↾_cat cresc 17761

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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-resc 17764

This theorem is referenced by:  rescbas  17782  rescbasOLD  17783  reschom  17784  rescco  17786  resccoOLD  17787  rescabs  17788  rescabsOLD  17789  rescabs2  17790  dfrngc2  20521  dfringc2  20550  rngcresringcat  20562  rngcrescrhm  20577  rngcrescrhmALTV  47212