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Mirrors > Home > MPE Home > Th. List > fullresc | Structured version Visualization version GIF version |
Description: The category formed by structure restriction is the same as the category restriction. (Contributed by Mario Carneiro, 5-Jan-2017.) |
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))) |
Ref | Expression |
---|---|
fullresc | ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fullsubc.h | . . . . . 6 ⊢ 𝐻 = (Homf ‘𝐶) | |
2 | fullsubc.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
3 | eqid 2821 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | fullsubc.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
5 | 4 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ 𝐵) |
6 | simprl 769 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) | |
7 | 5, 6 | sseldd 3968 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵) |
8 | simprr 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) | |
9 | 5, 8 | sseldd 3968 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵) |
10 | 1, 2, 3, 7, 9 | homfval 16962 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦)) |
11 | 6, 8 | ovresd 7315 | . . . . . 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 16963 | . . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵) |
15 | xpss12 5570 | . . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) | |
16 | 4, 4, 15 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) |
17 | fnssres 6470 | . . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) | |
18 | 14, 16, 17 | sylancr 589 | . . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) |
19 | 12, 2, 13, 18, 4 | reschom 17100 | . . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸)) |
20 | 19 | oveqdr 7184 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
21 | 11, 20 | eqtr3d 2858 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
22 | fullsubc.d | . . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆) | |
23 | 22, 2 | ressbas2 16555 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷)) |
24 | 4, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
25 | fvex 6683 | . . . . . . . 8 ⊢ (Base‘𝐷) ∈ V | |
26 | 24, 25 | eqeltrdi 2921 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V) |
27 | 22, 3 | resshom 16691 | . . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷)) |
28 | 26, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) |
29 | 28 | oveqdr 7184 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
30 | 10, 21, 29 | 3eqtr3rd 2865 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
31 | 30 | ralrimivva 3191 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
32 | eqid 2821 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
33 | eqid 2821 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
34 | 12, 2, 13, 18, 4 | rescbas 17099 | . . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸)) |
35 | 32, 33, 24, 34 | homfeq 16964 | . . 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 2821 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
38 | 22, 37 | ressco 16692 | . . . . 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 17102 | . . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸)) |
41 | 39, 40 | eqtr3d 2858 | . . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸)) |
42 | 41, 36 | comfeqd 16977 | . 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸)) |
43 | 36, 42 | jca 514 | 1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 × cxp 5553 ↾ cres 5557 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 Hom chom 16576 compcco 16577 Catccat 16935 Homf chomf 16937 compfccomf 16938 ↾cat cresc 17078 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-hom 16589 df-cco 16590 df-homf 16941 df-comf 16942 df-resc 17081 |
This theorem is referenced by: resscat 17122 funcres2c 17171 ressffth 17208 funcsetcres2 17353 |
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