Theorem fullresc 17946

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 2762	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		fullsubc.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
5	4	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ 𝐵)
6		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
7	5, 6	sseldd 3935	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
8		simprr 785	. . . . . . 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 17786	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
11	6, 8	ovresd 7584	. . . . . 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 17787	. . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵)
15		xpss12 5674	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
16	4, 4, 15	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17		fnssres 6659	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
18	14, 16, 17	sylancr 599	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
19	12, 2, 13, 18, 4	reschom 17925	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
20	19	oveqdr 7445	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
21	11, 20	eqtr3d 2799	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22		fullsubc.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
23	22, 2	ressbas2 17336	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷))
24	4, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
25		fvex 6895	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
26	24, 25	eqeltrdi 2870	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
27	22, 3	resshom 17509	. . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
28	26, 27	syl 18	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
29	28	oveqdr 7445	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
30	10, 21, 29	3eqtr3rd 2806	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
31	30	ralrimivva 3207	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32		eqid 2762	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
33		eqid 2762	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
34	12, 2, 13, 18, 4	rescbas 17924	. . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸))
35	32, 33, 24, 34	homfeq 17788	. . 3 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
36	31, 35	mpbird 260	. 2 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐸))
37		eqid 2762	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
38	22, 37	ressco 17510	. . . . 5 ⊢ (𝑆 ∈ V → (comp‘𝐶) = (comp‘𝐷))
39	26, 38	syl 18	. . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷))
40	12, 2, 13, 18, 4, 37	rescco 17927	. . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸))
41	39, 40	eqtr3d 2799	. . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸))
42	41, 36	comfeqd 17801	. 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸))
43	36, 42	jca 521	1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ⊆ wss 3902   × cxp 5657   ↾ cres 5661   Fn wfn 6532  ‘cfv 6537  (class class class)co 7417  Basecbs 17307   ↾_s cress 17328  Hom chom 17359  compcco 17360  Catccat 17758  Hom_f chomf 17760  comp_fccomf 17761   ↾_cat cresc 17903

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8957  df-dom 8958  df-sdom 8959  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-2 12331  df-3 12332  df-4 12333  df-5 12334  df-6 12335  df-7 12336  df-8 12337  df-9 12338  df-n0 12533  df-z 12620  df-dec 12741  df-sets 17262  df-slot 17280  df-ndx 17292  df-base 17308  df-ress 17329  df-hom 17372  df-cco 17373  df-homf 17764  df-comf 17765  df-resc 17906

This theorem is used by:  resscat  17947  funcres2c  17998  ressffth  18035  funcsetcres2  18188