Theorem fullresc 17867

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 2761	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		fullsubc.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵)
5	4	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ 𝐵)
6		simprl 780	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆)
7	5, 6	sseldd 3937	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵)
8		simprr 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆)
9	5, 8	sseldd 3937	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵)
10	1, 2, 3, 7, 9	homfval 17707	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦))
11	6, 8	ovresd 7559	. . . . . 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 17708	. . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵)
15		xpss12 5660	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
16	4, 4, 15	syl2anc 593	. . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵))
17		fnssres 6640	. . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
18	14, 16, 17	sylancr 596	. . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆))
19	12, 2, 13, 18, 4	reschom 17846	. . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸))
20	19	oveqdr 7420	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦))
21	11, 20	eqtr3d 2798	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦))
22		fullsubc.d	. . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆)
23	22, 2	ressbas2 17257	. . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷))
24	4, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷))
25		fvex 6876	. . . . . . . 8 ⊢ (Base‘𝐷) ∈ V
26	24, 25	eqeltrdi 2869	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V)
27	22, 3	resshom 17430	. . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷))
28	26, 27	syl 17	. . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷))
29	28	oveqdr 7420	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦))
30	10, 21, 29	3eqtr3rd 2805	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
31	30	ralrimivva 3204	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))
32		eqid 2761	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
33		eqid 2761	. . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸)
34	12, 2, 13, 18, 4	rescbas 17845	. . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸))
35	32, 33, 24, 34	homfeq 17709	. . 3 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)))
36	31, 35	mpbird 259	. 2 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐸))
37		eqid 2761	. . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶)
38	22, 37	ressco 17431	. . . . 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 17848	. . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸))
41	39, 40	eqtr3d 2798	. . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸))
42	41, 36	comfeqd 17722	. 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸))
43	36, 42	jca 519	1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3904   × cxp 5643   ↾ cres 5647   Fn wfn 6512  ‘cfv 6517  (class class class)co 7392  Basecbs 17228   ↾_s cress 17249  Hom chom 17280  compcco 17281  Catccat 17679  Hom_f chomf 17681  comp_fccomf 17682   ↾_cat cresc 17824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-er 8673  df-en 8924  df-dom 8925  df-sdom 8926  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-9 12284  df-n0 12479  df-z 12566  df-dec 12686  df-sets 17183  df-slot 17201  df-ndx 17213  df-base 17229  df-ress 17250  df-hom 17293  df-cco 17294  df-homf 17685  df-comf 17686  df-resc 17827

This theorem is referenced by:  resscat  17868  funcres2c  17919  ressffth  17956  funcsetcres2  18109