Theorem fullresc 17896

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 2737	. . . . . 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 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
7	5, 6	sseldd 3984	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
8		simprr 773	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
9	5, 8	sseldd 3984	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵)
10	1, 2, 3, 7, 9	homfval 17735	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
11	6, 8	ovresd 7600	. . . . . 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 17736	. . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵)
15		xpss12 5700	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
16	4, 4, 15	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17		fnssres 6691	. . . . . . . . 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 17874	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
20	19	oveqdr 7459	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
21	11, 20	eqtr3d 2779	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22		fullsubc.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
23	22, 2	ressbas2 17283	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷))
24	4, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
25		fvex 6919	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
26	24, 25	eqeltrdi 2849	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
27	22, 3	resshom 17463	. . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
28	26, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
29	28	oveqdr 7459	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
30	10, 21, 29	3eqtr3rd 2786	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
31	30	ralrimivva 3202	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32		eqid 2737	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
33		eqid 2737	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
34	12, 2, 13, 18, 4	rescbas 17873	. . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸))
35	32, 33, 24, 34	homfeq 17737	. . 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 2737	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
38	22, 37	ressco 17464	. . . . 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 17876	. . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸))
41	39, 40	eqtr3d 2779	. . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸))
42	41, 36	comfeqd 17750	. 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 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ⊆ wss 3951   × cxp 5683   ↾ cres 5687   Fn wfn 6556  ‘cfv 6561  (class class class)co 7431  Basecbs 17247   ↾_s cress 17274  Hom chom 17308  compcco 17309  Catccat 17707  Hom_f chomf 17709  comp_fccomf 17710   ↾_cat cresc 17852

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-hom 17321  df-cco 17322  df-homf 17713  df-comf 17714  df-resc 17855

This theorem is referenced by:  resscat  17897  funcres2c  17948  ressffth  17985  funcsetcres2  18138