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| Mirrors > Home > MPE Home > Th. List > subcrcl | Structured version Visualization version GIF version | ||
| Description: Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.) |
| Ref | Expression |
|---|---|
| subcrcl | ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-subc 17856 | . 2 ⊢ Subcat = (𝑐 ∈ Cat ↦ {ℎ ∣ (ℎ ⊆cat (Homf ‘𝑐) ∧ [dom dom ℎ / 𝑠]∀𝑥 ∈ 𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥ℎ𝑥) ∧ ∀𝑦 ∈ 𝑠 ∀𝑧 ∈ 𝑠 ∀𝑓 ∈ (𝑥ℎ𝑦)∀𝑔 ∈ (𝑦ℎ𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝑐)𝑧)𝑓) ∈ (𝑥ℎ𝑧)))}) | |
| 2 | 1 | mptrcl 7025 | 1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 {cab 2714 ∀wral 3061 [wsbc 3788 〈cop 4632 class class class wbr 5143 dom cdm 5685 ‘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-sep 5296 ax-nul 5306 ax-pr 5432 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fv 6569 df-subc 17856 |
| This theorem is referenced by: subcssc 17885 subcidcl 17889 subccocl 17890 subccatid 17891 subsubc 17898 funcres2b 17942 funcres2 17943 idfusubc 17945 |
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