Theorem fullresc 17758

Description: The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.)

Hypotheses

Ref	Expression
fullsubc.b	⊢ 𝐵 = (Base‘𝐶)
fullsubc.h	⊢ 𝐻 = (Homf ‘𝐶)
fullsubc.c	⊢ (𝜑 → 𝐶 ∈ Cat)
fullsubc.s	⊢ (𝜑 → 𝑆 ⊆ 𝐵)
fullsubc.d	⊢ 𝐷 = (𝐶 ↾s 𝑆)
fullsubc.e	⊢ 𝐸 = (𝐶 ↾cat (𝐻 ↾ (𝑆 × 𝑆)))

Assertion

Ref	Expression
fullresc	⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸)))

Proof of Theorem fullresc

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fullsubc.h	. . . . . 6 ⊢ 𝐻 = (Homf ‘𝐶)
2		fullsubc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2731	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		fullsubc.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
5	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ 𝐵)
6		simprl 770	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
7	5, 6	sseldd 3935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
8		simprr 772	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
9	5, 8	sseldd 3935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵)
10	1, 2, 3, 7, 9	homfval 17598	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
11	6, 8	ovresd 7513	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥𝐻𝑦))
12		fullsubc.e	. . . . . . . 8 ⊢ 𝐸 = (𝐶 ↾cat (𝐻 ↾ (𝑆 × 𝑆)))
13		fullsubc.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ Cat)
14	1, 2	homffn 17599	. . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵)
15		xpss12 5631	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
16	4, 4, 15	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17		fnssres 6604	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
18	14, 16, 17	sylancr 587	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
19	12, 2, 13, 18, 4	reschom 17737	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
20	19	oveqdr 7374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
21	11, 20	eqtr3d 2768	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22		fullsubc.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
23	22, 2	ressbas2 17149	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷))
24	4, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
25		fvex 6835	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
26	24, 25	eqeltrdi 2839	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
27	22, 3	resshom 17322	. . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
28	26, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
29	28	oveqdr 7374	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
30	10, 21, 29	3eqtr3rd 2775	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
31	30	ralrimivva 3175	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32		eqid 2731	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
33		eqid 2731	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
34	12, 2, 13, 18, 4	rescbas 17736	. . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸))
35	32, 33, 24, 34	homfeq 17600	. . 3 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
36	31, 35	mpbird 257	. 2 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐸))
37		eqid 2731	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
38	22, 37	ressco 17323	. . . . 5 ⊢ (𝑆 ∈ V → (comp‘𝐶) = (comp‘𝐷))
39	26, 38	syl 17	. . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷))
40	12, 2, 13, 18, 4, 37	rescco 17739	. . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸))
41	39, 40	eqtr3d 2768	. . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸))
42	41, 36	comfeqd 17613	. 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸))
43	36, 42	jca 511	1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3902   × cxp 5614   ↾ cres 5618   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346  Basecbs 17120   ↾_s cress 17141  Hom chom 17172  compcco 17173  Catccat 17570  Hom_f chomf 17572  comp_fccomf 17573   ↾_cat cresc 17715

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-z 12469  df-dec 12589  df-sets 17075  df-slot 17093  df-ndx 17105  df-base 17121  df-ress 17142  df-hom 17185  df-cco 17186  df-homf 17576  df-comf 17577  df-resc 17718

This theorem is referenced by:  resscat  17759  funcres2c  17810  ressffth  17847  funcsetcres2  18000