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Mirrors > Home > MPE Home > Th. List > initoo | Structured version Visualization version GIF version |
Description: An initial object is an object. (Contributed by AV, 14-Apr-2020.) |
Ref | Expression |
---|---|
initoo | ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
2 | eqid 2818 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | id 22 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
4 | 1, 2, 3 | isinitoi 17251 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑂(Hom ‘𝐶)𝑏))) |
5 | 4 | simpld 495 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)) |
6 | 5 | ex 413 | 1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ (Base‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∃!weu 2646 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 Hom chom 16564 Catccat 16923 InitOcinito 17236 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-inito 17239 |
This theorem is referenced by: iszeroi 17257 2initoinv 17258 initoeu1w 17260 |
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