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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 2725 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | fullsubc.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
5 | 4 | adantr 479 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ 𝐵) |
6 | simprl 769 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) | |
7 | 5, 6 | sseldd 3977 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵) |
8 | simprr 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) | |
9 | 5, 8 | sseldd 3977 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵) |
10 | 1, 2, 3, 7, 9 | homfval 17675 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦)) |
11 | 6, 8 | ovresd 7588 | . . . . . 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 17676 | . . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵) |
15 | xpss12 5693 | . . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) | |
16 | 4, 4, 15 | syl2anc 582 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) |
17 | fnssres 6679 | . . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) | |
18 | 14, 16, 17 | sylancr 585 | . . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) |
19 | 12, 2, 13, 18, 4 | reschom 17817 | . . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸)) |
20 | 19 | oveqdr 7447 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
21 | 11, 20 | eqtr3d 2767 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
22 | fullsubc.d | . . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆) | |
23 | 22, 2 | ressbas2 17221 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷)) |
24 | 4, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
25 | fvex 6909 | . . . . . . . 8 ⊢ (Base‘𝐷) ∈ V | |
26 | 24, 25 | eqeltrdi 2833 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V) |
27 | 22, 3 | resshom 17403 | . . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷)) |
28 | 26, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) |
29 | 28 | oveqdr 7447 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
30 | 10, 21, 29 | 3eqtr3rd 2774 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
31 | 30 | ralrimivva 3190 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
32 | eqid 2725 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
33 | eqid 2725 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
34 | 12, 2, 13, 18, 4 | rescbas 17815 | . . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸)) |
35 | 32, 33, 24, 34 | homfeq 17677 | . . 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 2725 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
38 | 22, 37 | ressco 17404 | . . . . 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 17819 | . . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸)) |
41 | 39, 40 | eqtr3d 2767 | . . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸)) |
42 | 41, 36 | comfeqd 17690 | . 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸)) |
43 | 36, 42 | jca 510 | 1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ⊆ wss 3944 × cxp 5676 ↾ cres 5680 Fn wfn 6544 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 ↾s cress 17212 Hom chom 17247 compcco 17248 Catccat 17647 Homf chomf 17649 compfccomf 17650 ↾cat cresc 17794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-hom 17260 df-cco 17261 df-homf 17653 df-comf 17654 df-resc 17797 |
This theorem is referenced by: resscat 17841 funcres2c 17893 ressffth 17930 funcsetcres2 18085 |
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