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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 16472 | . . 3 ⊢ 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 | dmmptss 5631 | . 2 ⊢ dom Subcat ⊆ Cat |
3 | elfvdm 6220 | . 2 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ dom Subcat) | |
4 | 2, 3 | sseldi 3601 | 1 ⊢ (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 {cab 2608 ∀wral 2912 [wsbc 3435 〈cop 4183 class class class wbr 4653 dom cdm 5114 ‘cfv 5888 (class class class)co 6650 compcco 15953 Catccat 16325 Idccid 16326 Homf chomf 16327 ⊆cat cssc 16467 Subcatcsubc 16469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fv 5896 df-subc 16472 |
This theorem is referenced by: subcssc 16500 subcidcl 16504 subccocl 16505 subccatid 16506 subsubc 16513 funcres2b 16557 funcres2 16558 idfusubc 41866 |
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