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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 6437 | . . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
3 | 2, 2 | oveq12d 6928 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋)) |
4 | 1, 3 | eleq12d 2900 | . 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))) |
5 | subcidcl.j | . . . . 5 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
6 | eqid 2825 | . . . . . 6 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
7 | subcidcl.1 | . . . . . 6 ⊢ 1 = (Id‘𝐶) | |
8 | eqid 2825 | . . . . . 6 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | subcrcl 16835 | . . . . . . 7 ⊢ (𝐽 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) | |
10 | 5, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) |
11 | subcidcl.2 | . . . . . 6 ⊢ (𝜑 → 𝐽 Fn (𝑆 × 𝑆)) | |
12 | 6, 7, 8, 10, 11 | issubc2 16855 | . . . . 5 ⊢ (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 224 | . . . 4 ⊢ (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 | 13 | simprd 491 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧))) |
15 | simpl 476 | . . . 4 ⊢ ((( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ∧ ∀𝑦 ∈ 𝑆 ∀𝑧 ∈ 𝑆 ∀𝑓 ∈ (𝑥𝐽𝑦)∀𝑔 ∈ (𝑦𝐽𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐽𝑧)) → ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) | |
16 | 15 | ralimi 3161 | . . 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)) |
17 | 14, 16 | syl 17 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) |
18 | subcidcl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
19 | 4, 17, 18 | rspcdva 3532 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∀wral 3117 〈cop 4405 class class class wbr 4875 × cxp 5344 Fn wfn 6122 ‘cfv 6127 (class class class)co 6910 compcco 16324 Catccat 16684 Idccid 16685 Homf chomf 16686 ⊆cat cssc 16826 Subcatcsubc 16828 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-pm 8130 df-ixp 8182 df-ssc 16829 df-subc 16831 |
This theorem is referenced by: subccatid 16865 issubc3 16868 funcres 16915 |
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