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Mirrors > Home > MPE Home > Th. List > initorcl | Structured version Visualization version GIF version |
Description: Reverse closure for an initial object: If a class has an initial object, the class is a category. (Contributed by AV, 4-Apr-2020.) |
Ref | Expression |
---|---|
initorcl | ⊢ (𝐼 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inito 17245 | . 2 ⊢ InitO = (𝑐 ∈ Cat ↦ {𝑎 ∈ (Base‘𝑐) ∣ ∀𝑏 ∈ (Base‘𝑐)∃!ℎ ℎ ∈ (𝑎(Hom ‘𝑐)𝑏)}) | |
2 | 1 | mptrcl 6771 | 1 ⊢ (𝐼 ∈ (InitO‘𝐶) → 𝐶 ∈ Cat) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∃!weu 2649 ∀wral 3138 {crab 3142 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Catccat 16929 InitOcinito 17242 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-xp 5555 df-rel 5556 df-cnv 5557 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fv 6357 df-inito 17245 |
This theorem is referenced by: (None) |
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