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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 17082 | . 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 6777 | 1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 {cab 2799 ∀wral 3138 [wsbc 3772 〈cop 4573 class class class wbr 5066 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 compcco 16577 Catccat 16935 Idccid 16936 Homf chomf 16937 ⊆cat cssc 17077 Subcatcsubc 17079 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fv 6363 df-subc 17082 |
This theorem is referenced by: subcssc 17110 subcidcl 17114 subccocl 17115 subccatid 17116 subsubc 17123 funcres2b 17167 funcres2 17168 idfusubc 44186 |
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