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Mirrors > Home > MPE Home > Th. List > isinitoi | Structured version Visualization version GIF version |
Description: Implication of a class being an initial object. (Contributed by AV, 6-Apr-2020.) |
Ref | Expression |
---|---|
isinitoi.b | ⊢ 𝐵 = (Base‘𝐶) |
isinitoi.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isinitoi.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
Ref | Expression |
---|---|
isinitoi | ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinitoi.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | isinitoi.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐶) | |
3 | isinitoi.h | . . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | initoval 17249 | . . . . 5 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)}) |
5 | 4 | eleq2d 2875 | . . . 4 ⊢ (𝜑 → (𝑂 ∈ (InitO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})) |
6 | elrabi 3623 | . . . 4 ⊢ (𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)} → 𝑂 ∈ 𝐵) | |
7 | 5, 6 | syl6bi 256 | . . 3 ⊢ (𝜑 → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵)) |
8 | 7 | imp 410 | . 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ 𝐵) |
9 | 1 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat) |
10 | simpr 488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵) | |
11 | 2, 3, 9, 10 | isinito 17252 | . . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
12 | 11 | biimpd 232 | . . 3 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (InitO‘𝐶) → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
13 | 12 | impancom 455 | . 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
14 | 8, 13 | jcai 520 | 1 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃!weu 2628 ∀wral 3106 {crab 3110 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 Hom chom 16568 Catccat 16927 InitOcinito 17240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-inito 17243 |
This theorem is referenced by: initoid 17257 initoo 17259 initoeu1 17263 initoeu2 17268 |
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