Theorem rescval2 17790

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 7727	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
5		fnex 7191	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
6	2, 4, 5	syl2anc 584	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
7		rescval.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
8	7	rescval 17789	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
9	1, 6, 8	syl2anc 584	. 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
10	2	fndmd 6623	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
11	10	dmeqd 5869	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12		dmxpid 5894	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
13	11, 12	eqtrdi 2780	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
14	13	oveq2d 7403	. . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆))
15	14	oveq1d 7402	. 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
16	9, 15	eqtrd 2764	1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3447  ⟨cop 4595   × cxp 5636  dom cdm 5638   Fn wfn 6506  ‘cfv 6511  (class class class)co 7387   sSet csts 17133  ndxcnx 17163   ↾_s cress 17200  Hom chom 17231   ↾_cat cresc 17770

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-resc 17773

This theorem is referenced by:  rescbas  17791  reschom  17792  rescco  17794  rescabs  17795  rescabs2  17796  dfrngc2  20537  dfringc2  20566  rngcresringcat  20578  rngcrescrhm  20593  rngcrescrhmALTV  48268