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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 2798 | . . . . . 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 770 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) | |
7 | 5, 6 | sseldd 3916 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵) |
8 | simprr 772 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) | |
9 | 5, 8 | sseldd 3916 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵) |
10 | 1, 2, 3, 7, 9 | homfval 16954 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦)) |
11 | 6, 8 | ovresd 7295 | . . . . . 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 16955 | . . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵) |
15 | xpss12 5534 | . . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) | |
16 | 4, 4, 15 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) |
17 | fnssres 6442 | . . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) | |
18 | 14, 16, 17 | sylancr 590 | . . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) |
19 | 12, 2, 13, 18, 4 | reschom 17092 | . . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸)) |
20 | 19 | oveqdr 7163 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
21 | 11, 20 | eqtr3d 2835 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
22 | fullsubc.d | . . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆) | |
23 | 22, 2 | ressbas2 16547 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷)) |
24 | 4, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
25 | fvex 6658 | . . . . . . . 8 ⊢ (Base‘𝐷) ∈ V | |
26 | 24, 25 | eqeltrdi 2898 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V) |
27 | 22, 3 | resshom 16683 | . . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷)) |
28 | 26, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) |
29 | 28 | oveqdr 7163 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
30 | 10, 21, 29 | 3eqtr3rd 2842 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
31 | 30 | ralrimivva 3156 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
32 | eqid 2798 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
33 | eqid 2798 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
34 | 12, 2, 13, 18, 4 | rescbas 17091 | . . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸)) |
35 | 32, 33, 24, 34 | homfeq 16956 | . . 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 2798 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
38 | 22, 37 | ressco 16684 | . . . . 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 17094 | . . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸)) |
41 | 39, 40 | eqtr3d 2835 | . . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸)) |
42 | 41, 36 | comfeqd 16969 | . 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸)) |
43 | 36, 42 | jca 515 | 1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 × cxp 5517 ↾ cres 5521 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 Hom chom 16568 compcco 16569 Catccat 16927 Homf chomf 16929 compfccomf 16930 ↾cat cresc 17070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-hom 16581 df-cco 16582 df-homf 16933 df-comf 16934 df-resc 17073 |
This theorem is referenced by: resscat 17114 funcres2c 17163 ressffth 17200 funcsetcres2 17345 |
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