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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 17084 | . 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 6779 | 1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 {cab 2801 ∀wral 3140 [wsbc 3774 〈cop 4575 class class class wbr 5068 dom cdm 5557 ‘cfv 6357 (class class class)co 7158 compcco 16579 Catccat 16937 Idccid 16938 Homf chomf 16939 ⊆cat cssc 17079 Subcatcsubc 17081 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-xp 5563 df-rel 5564 df-cnv 5565 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fv 6365 df-subc 17084 |
This theorem is referenced by: subcssc 17112 subcidcl 17116 subccocl 17117 subccatid 17118 subsubc 17125 funcres2b 17169 funcres2 17170 idfusubc 44144 |
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