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Mirrors > Home > MPE Home > Th. List > subcidcl | Structured version Visualization version GIF version |
Description: The identity of the original category is contained in each subcategory. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
subcidcl.j | ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) |
subcidcl.2 | ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) |
subcidcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
subcidcl.1 | ⊢ 1 = (Id‘𝐶) |
Ref | Expression |
---|---|
subcidcl | ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6645 | . . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
3 | 2, 2 | oveq12d 7153 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋)) |
4 | 1, 3 | eleq12d 2884 | . 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))) |
5 | subcidcl.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
6 | eqid 2798 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
7 | subcidcl.1 | . . . . 5 ⊢ 1 = (Id‘𝐶) | |
8 | eqid 2798 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | subcrcl 17078 | . . . . . 6 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) | |
10 | 5, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Cat) |
11 | subcidcl.2 | . . . . 5 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
12 | 6, 7, 8, 10, 11 | issubc2 17098 | . . . 4 ⊢ (𝜑 → (𝐽 ∈ (Subcat‘𝐶) ↔ (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))))) |
13 | 5, 12 | mpbid 235 | . . 3 ⊢ (𝜑 → (𝐽 ⊆cat (Homf ‘𝐶) ∧ ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)))) |
14 | simpl 486 | . . . 4 ⊢ ((( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) | |
15 | 14 | ralimi 3128 | . . 3 ⊢ (∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) |
16 | 13, 15 | simpl2im 507 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) |
17 | subcidcl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
18 | 4, 16, 17 | rspcdva 3573 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 class class class wbr 5030 × cxp 5517 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 compcco 16569 Catccat 16927 Idccid 16928 Homf chomf 16929 ⊆cat cssc 17069 Subcatcsubc 17071 |
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 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 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-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-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 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-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pm 8392 df-ixp 8445 df-ssc 17072 df-subc 17074 |
This theorem is referenced by: subccatid 17108 issubc3 17111 funcres 17158 |
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