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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 2737 | . . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 4 | fullsubc.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
| 5 | 4 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑆 ⊆ 𝐵) |
| 6 | simprl 771 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝑆) | |
| 7 | 5, 6 | sseldd 3984 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑥 ∈ 𝐵) |
| 8 | simprr 773 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝑆) | |
| 9 | 5, 8 | sseldd 3984 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → 𝑦 ∈ 𝐵) |
| 10 | 1, 2, 3, 7, 9 | homfval 17735 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐶)𝑦)) |
| 11 | 6, 8 | ovresd 7600 | . . . . . 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 17736 | . . . . . . . . 9 ⊢ 𝐻 Fn (𝐵 × 𝐵) |
| 15 | xpss12 5700 | . . . . . . . . . 10 ⊢ ((𝑆 ⊆ 𝐵 ∧ 𝑆 ⊆ 𝐵) → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) | |
| 16 | 4, 4, 15 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝑆 × 𝑆) ⊆ (𝐵 × 𝐵)) |
| 17 | fnssres 6691 | . . . . . . . . 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 17874 | . . . . . . 7 ⊢ (𝜑 → (𝐻 ↾ (𝑆 × 𝑆)) = (Hom ‘𝐸)) |
| 20 | 19 | oveqdr 7459 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(𝐻 ↾ (𝑆 × 𝑆))𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
| 21 | 11, 20 | eqtr3d 2779 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐻𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
| 22 | fullsubc.d | . . . . . . . . . 10 ⊢ 𝐷 = (𝐶 ↾s 𝑆) | |
| 23 | 22, 2 | ressbas2 17283 | . . . . . . . . 9 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 = (Base‘𝐷)) |
| 24 | 4, 23 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = (Base‘𝐷)) |
| 25 | fvex 6919 | . . . . . . . 8 ⊢ (Base‘𝐷) ∈ V | |
| 26 | 24, 25 | eqeltrdi 2849 | . . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ V) |
| 27 | 22, 3 | resshom 17463 | . . . . . . 7 ⊢ (𝑆 ∈ V → (Hom ‘𝐶) = (Hom ‘𝐷)) |
| 28 | 26, 27 | syl 17 | . . . . . 6 ⊢ (𝜑 → (Hom ‘𝐶) = (Hom ‘𝐷)) |
| 29 | 28 | oveqdr 7459 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐶)𝑦) = (𝑥(Hom ‘𝐷)𝑦)) |
| 30 | 10, 21, 29 | 3eqtr3rd 2786 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
| 31 | 30 | ralrimivva 3202 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦)) |
| 32 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷) | |
| 33 | eqid 2737 | . . . 4 ⊢ (Hom ‘𝐸) = (Hom ‘𝐸) | |
| 34 | 12, 2, 13, 18, 4 | rescbas 17873 | . . . 4 ⊢ (𝜑 → 𝑆 = (Base‘𝐸)) |
| 35 | 32, 33, 24, 34 | homfeq 17737 | . . 3 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(Hom ‘𝐷)𝑦) = (𝑥(Hom ‘𝐸)𝑦))) |
| 36 | 31, 35 | mpbird 257 | . 2 ⊢ (𝜑 → (Homf ‘𝐷) = (Homf ‘𝐸)) |
| 37 | eqid 2737 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 38 | 22, 37 | ressco 17464 | . . . . 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 17876 | . . . 4 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐸)) |
| 41 | 39, 40 | eqtr3d 2779 | . . 3 ⊢ (𝜑 → (comp‘𝐷) = (comp‘𝐸)) |
| 42 | 41, 36 | comfeqd 17750 | . 2 ⊢ (𝜑 → (compf‘𝐷) = (compf‘𝐸)) |
| 43 | 36, 42 | jca 511 | 1 ⊢ (𝜑 → ((Homf ‘𝐷) = (Homf ‘𝐸) ∧ (compf‘𝐷) = (compf‘𝐸))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 × cxp 5683 ↾ cres 5687 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 Hom chom 17308 compcco 17309 Catccat 17707 Homf chomf 17709 compfccomf 17710 ↾cat cresc 17852 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-hom 17321 df-cco 17322 df-homf 17713 df-comf 17714 df-resc 17855 |
| This theorem is referenced by: resscat 17897 funcres2c 17948 ressffth 17985 funcsetcres2 18138 |
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