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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 6651 | . . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋)) | |
2 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
3 | 2, 2 | oveq12d 7161 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋)) |
4 | 1, 3 | eleq12d 2845 | . 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))) |
5 | subcidcl.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
6 | eqid 2759 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
7 | subcidcl.1 | . . . . 5 ⊢ 1 = (Id‘𝐶) | |
8 | eqid 2759 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
9 | subcrcl 17130 | . . . . . 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 17150 | . . . 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 487 | . . . 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 3090 | . . 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 508 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) |
17 | subcidcl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
18 | 4, 16, 17 | rspcdva 3541 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 400 = wceq 1539 ∈ wcel 2112 ∀wral 3068 〈cop 4521 class class class wbr 5025 × cxp 5515 Fn wfn 6323 ‘cfv 6328 (class class class)co 7143 compcco 16620 Catccat 16978 Idccid 16979 Homf chomf 16980 ⊆cat cssc 17121 Subcatcsubc 17123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5149 ax-sep 5162 ax-nul 5169 ax-pow 5227 ax-pr 5291 ax-un 7452 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-reu 3075 df-rab 3077 df-v 3409 df-sbc 3694 df-csb 3802 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-pw 4489 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-iun 4878 df-br 5026 df-opab 5088 df-mpt 5106 df-id 5423 df-xp 5523 df-rel 5524 df-cnv 5525 df-co 5526 df-dm 5527 df-rn 5528 df-res 5529 df-ima 5530 df-iota 6287 df-fun 6330 df-fn 6331 df-f 6332 df-f1 6333 df-fo 6334 df-f1o 6335 df-fv 6336 df-ov 7146 df-oprab 7147 df-mpo 7148 df-pm 8412 df-ixp 8473 df-ssc 17124 df-subc 17126 |
This theorem is referenced by: subccatid 17160 issubc3 17163 funcres 17210 |
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