![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > reschom | Structured version Visualization version GIF version |
Description: Hom-sets of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rescbas.d | ⊢ 𝐷 = (𝐶 ↾cat 𝐻) |
rescbas.b | ⊢ 𝐵 = (Base‘𝐶) |
rescbas.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
rescbas.h | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
rescbas.s | ⊢ (𝜑 → 𝑆 ⊆ 𝐵) |
Ref | Expression |
---|---|
reschom | ⊢ (𝜑 → 𝐻 = (Hom ‘𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ovex 7449 | . . 3 ⊢ (𝐶 ↾s 𝑆) ∈ V | |
2 | rescbas.h | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
3 | rescbas.s | . . . . . 6 ⊢ (𝜑 → 𝑆 ⊆ 𝐵) | |
4 | rescbas.b | . . . . . . . 8 ⊢ 𝐵 = (Base‘𝐶) | |
5 | 4 | fvexi 6907 | . . . . . . 7 ⊢ 𝐵 ∈ V |
6 | 5 | ssex 5318 | . . . . . 6 ⊢ (𝑆 ⊆ 𝐵 → 𝑆 ∈ V) |
7 | 3, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ V) |
8 | 7, 7 | xpexd 7751 | . . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V) |
9 | fnex 7226 | . . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
10 | 2, 8, 9 | syl2anc 582 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
11 | homid 17421 | . . . 4 ⊢ Hom = Slot (Hom ‘ndx) | |
12 | 11 | setsid 17205 | . . 3 ⊢ (((𝐶 ↾s 𝑆) ∈ V ∧ 𝐻 ∈ V) → 𝐻 = (Hom ‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
13 | 1, 10, 12 | sylancr 585 | . 2 ⊢ (𝜑 → 𝐻 = (Hom ‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
14 | rescbas.d | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
15 | rescbas.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
16 | 14, 15, 7, 2 | rescval2 17839 | . . 3 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
17 | 16 | fveq2d 6897 | . 2 ⊢ (𝜑 → (Hom ‘𝐷) = (Hom ‘((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉))) |
18 | 13, 17 | eqtr4d 2769 | 1 ⊢ (𝜑 → 𝐻 = (Hom ‘𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⊆ wss 3946 〈cop 4629 × cxp 5672 Fn wfn 6541 ‘cfv 6546 (class class class)co 7416 sSet csts 17160 ndxcnx 17190 Basecbs 17208 ↾s cress 17237 Hom chom 17272 ↾cat cresc 17819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-ltxr 11294 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-dec 12724 df-sets 17161 df-slot 17179 df-ndx 17191 df-hom 17285 df-resc 17822 |
This theorem is referenced by: reschomf 17843 subccatid 17860 issubc3 17863 fullresc 17865 funcres 17910 funcres2b 17911 funcres2 17912 idfusubc 17914 rngchomfval 20596 ringchomfval 20625 subthinc 48397 |
Copyright terms: Public domain | W3C validator |