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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 6900 | . . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
3 | 2, 2 | oveq12d 7441 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋)) |
4 | 1, 3 | eleq12d 2819 | . 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))) |
5 | subcidcl.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
6 | eqid 2725 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
7 | subcidcl.1 | . . . . 5 ⊢ 1 = (Id‘𝐶) | |
8 | eqid 2725 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | subcrcl 17827 | . . . . . 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 17850 | . . . 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 231 | . . 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 481 | . . . 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 3072 | . . 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 502 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) |
17 | subcidcl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
18 | 4, 16, 17 | rspcdva 3608 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 〈cop 4638 class class class wbr 5152 × cxp 5679 Fn wfn 6548 ‘cfv 6553 (class class class)co 7423 compcco 17273 Catccat 17672 Idccid 17673 Homf chomf 17674 ⊆cat cssc 17818 Subcatcsubc 17820 |
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 5289 ax-sep 5303 ax-nul 5310 ax-pow 5368 ax-pr 5432 ax-un 7745 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4325 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5579 df-xp 5687 df-rel 5688 df-cnv 5689 df-co 5690 df-dm 5691 df-rn 5692 df-res 5693 df-ima 5694 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7426 df-oprab 7427 df-mpo 7428 df-pm 8857 df-ixp 8926 df-ssc 17821 df-subc 17823 |
This theorem is referenced by: subccatid 17860 issubc3 17863 funcres 17910 |
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