Theorem rescval2 18053

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 7797	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
5		fnex 7254	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
6	2, 4, 5	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
7		rescval.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
8	7	rescval 18052	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
9	1, 6, 8	syl2anc 596	. 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
10	2	fndmd 6674	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
11	10	dmeqd 5899	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
12		dmxpid 5924	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
13	11, 12	eqtrdi 2814	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
14	13	oveq2d 7467	. . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆))
15	14	oveq1d 7466	. 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
16	9, 15	eqtrd 2798	1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ⟨cop 4840   × cxp 5656  dom cdm 5658   Fn wfn 6561  ‘cfv 6566  (class class class)co 7451   sSet csts 17385  ndxcnx 17415   ↾_s cress 17452  Hom chom 17483   ↾_cat cresc 18033

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-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-resc 18036

This theorem is used by:  rescbas  18054  reschom  18055  rescco  18057  rescabs  18058  rescabs2  18059  dfrngc2  20931  dfringc2  20960  rngcresringcat  20972  rngcrescrhm  20987  rngcrescrhmALTV  49650