Theorem fullresc 17482

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 2738	. . . . . 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 767	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
7	5, 6	sseldd 3918	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
8		simprr 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
9	5, 8	sseldd 3918	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵)
10	1, 2, 3, 7, 9	homfval 17318	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
11	6, 8	ovresd 7417	. . . . . 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 17319	. . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵)
15		xpss12 5595	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
16	4, 4, 15	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17		fnssres 6539	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
18	14, 16, 17	sylancr 586	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
19	12, 2, 13, 18, 4	reschom 17460	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
20	19	oveqdr 7283	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
21	11, 20	eqtr3d 2780	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22		fullsubc.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
23	22, 2	ressbas2 16875	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷))
24	4, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
25		fvex 6769	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
26	24, 25	eqeltrdi 2847	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
27	22, 3	resshom 17046	. . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
28	26, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
29	28	oveqdr 7283	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
30	10, 21, 29	3eqtr3rd 2787	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
31	30	ralrimivva 3114	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32		eqid 2738	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
33		eqid 2738	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
34	12, 2, 13, 18, 4	rescbas 17458	. . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸))
35	32, 33, 24, 34	homfeq 17320	. . 3 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
36	31, 35	mpbird 256	. 2 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐸))
37		eqid 2738	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
38	22, 37	ressco 17047	. . . . 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 17462	. . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸))
41	39, 40	eqtr3d 2780	. . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸))
42	41, 36	comfeqd 17333	. 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 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883   × cxp 5578   ↾ cres 5582   Fn wfn 6413  ‘cfv 6418  (class class class)co 7255  Basecbs 16840   ↾_s cress 16867  Hom chom 16899  compcco 16900  Catccat 17290  Hom_f chomf 17292  comp_fccomf 17293   ↾_cat cresc 17437

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-hom 16912  df-cco 16913  df-homf 17296  df-comf 17297  df-resc 17440

This theorem is referenced by:  resscat  17483  funcres2c  17533  ressffth  17570  funcsetcres2  17724