Theorem rescval2 17098

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 7474	. . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V)
5		fnex 6980	. . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V)
6	2, 4, 5	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐻 ∈ V)
7		rescval.1	. . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻)
8	7	rescval 17097	. . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
9	1, 6, 8	syl2anc 586	. 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩))
10		fndm 6455	. . . . . . 7 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆))
11	2, 10	syl 17	. . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆))
12	11	dmeqd 5774	. . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆))
13		dmxpid 5800	. . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆
14	12, 13	syl6eq 2872	. . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆)
15	14	oveq2d 7172	. . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆))
16	15	oveq1d 7171	. 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet ⟨(Hom ‘ndx), 𝐻⟩) = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))
17	9, 16	eqtrd 2856	1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ⟨cop 4573   × cxp 5553  dom cdm 5555   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156  ndxcnx 16480   sSet csts 16481   ↾_s cress 16484  Hom chom 16576   ↾_cat cresc 17078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-resc 17081

This theorem is referenced by:  rescbas  17099  reschom  17100  rescco  17102  rescabs  17103  rescabs2  17104  dfrngc2  44263  dfringc2  44309  rngcresringcat  44321  rngcrescrhm  44376  rngcrescrhmALTV  44394