Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2740 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
4 | fullsubc.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
5 | 4 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ 𝐵) |
6 | simprl 768 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) | |
7 | 5, 6 | sseldd 3927 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵) |
8 | simprr 770 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) | |
9 | 5, 8 | sseldd 3927 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵) |
10 | 1, 2, 3, 7, 9 | homfval 17399 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦)) |
11 | 6, 8 | ovresd 7433 | . . . . . 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 17400 | . . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵) |
15 | xpss12 5605 | . . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) | |
16 | 4, 4, 15 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) |
17 | fnssres 6553 | . . . . . . . . 9 ⊢ ((𝐻 Fn (𝐵 × 𝐵) ∧ (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) | |
18 | 14, 16, 17 | sylancr 587 | . . . . . . . 8 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) Fn (𝑆 × 𝑆)) |
19 | 12, 2, 13, 18, 4 | reschom 17541 | . . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸)) |
20 | 19 | oveqdr 7299 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
21 | 11, 20 | eqtr3d 2782 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
22 | fullsubc.d | . . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆) | |
23 | 22, 2 | ressbas2 16947 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷)) |
24 | 4, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
25 | fvex 6784 | . . . . . . . 8 ⊢ (Base‘𝐷) ∈ V | |
26 | 24, 25 | eqeltrdi 2849 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V) |
27 | 22, 3 | resshom 17127 | . . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷)) |
28 | 26, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) |
29 | 28 | oveqdr 7299 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
30 | 10, 21, 29 | 3eqtr3rd 2789 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
31 | 30 | ralrimivva 3117 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
32 | eqid 2740 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
33 | eqid 2740 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
34 | 12, 2, 13, 18, 4 | rescbas 17539 | . . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸)) |
35 | 32, 33, 24, 34 | homfeq 17401 | . . 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 2740 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
38 | 22, 37 | ressco 17128 | . . . . 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 17543 | . . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸)) |
41 | 39, 40 | eqtr3d 2782 | . . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸)) |
42 | 41, 36 | comfeqd 17414 | . 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸)) |
43 | 36, 42 | jca 512 | 1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 Vcvv 3431 ⊆ wss 3892 × cxp 5588 ↾ cres 5592 Fn wfn 6427 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 ↾s cress 16939 Hom chom 16971 compcco 16972 Catccat 17371 Homf chomf 17373 compfccomf 17374 ↾cat cresc 17518 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-om 7707 df-1st 7824 df-2nd 7825 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-er 8481 df-en 8717 df-dom 8718 df-sdom 8719 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-hom 16984 df-cco 16985 df-homf 17377 df-comf 17378 df-resc 17521 |
This theorem is referenced by: resscat 17565 funcres2c 17615 ressffth 17652 funcsetcres2 17806 |
Copyright terms: Public domain | W3C validator |