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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 6906 | . . 3 ⊢ (𝑥 = 𝑋 → ( 1 ‘𝑥) = ( 1 ‘𝑋)) | |
| 2 | id 22 | . . . 4 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋) | |
| 3 | 2, 2 | oveq12d 7449 | . . 3 ⊢ (𝑥 = 𝑋 → (𝑥𝐽𝑥) = (𝑋𝐽𝑋)) | 
| 4 | 1, 3 | eleq12d 2835 | . 2 ⊢ (𝑥 = 𝑋 → (( 1 ‘𝑥) ∈ (𝑥𝐽𝑥) ↔ ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋))) | 
| 5 | subcidcl.j | . . . 4 ⊢ (𝜑 → 𝐽 ∈ (Subcat‘𝐶)) | |
| 6 | eqid 2737 | . . . . 5 ⊢ (Homf ‘𝐶) = (Homf ‘𝐶) | |
| 7 | subcidcl.1 | . . . . 5 ⊢ 1 = (Id‘𝐶) | |
| 8 | eqid 2737 | . . . . 5 ⊢ (comp‘𝐶) = (comp‘𝐶) | |
| 9 | subcrcl 17860 | . . . . . 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 17881 | . . . 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 232 | . . 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 482 | . . . 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 3083 | . . 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 503 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 ( 1 ‘𝑥) ∈ (𝑥𝐽𝑥)) | 
| 17 | subcidcl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
| 18 | 4, 16, 17 | rspcdva 3623 | 1 ⊢ (𝜑 → ( 1 ‘𝑋) ∈ (𝑋𝐽𝑋)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 〈cop 4632 class class class wbr 5143 × cxp 5683 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 compcco 17309 Catccat 17707 Idccid 17708 Homf chomf 17709 ⊆cat cssc 17851 Subcatcsubc 17853 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8869 df-ixp 8938 df-ssc 17854 df-subc 17856 | 
| This theorem is referenced by: subccatid 17891 issubc3 17894 funcres 17941 | 
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