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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 18047 | . . . . 5 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)}) |
5 | 4 | eleq2d 2825 | . . . 4 ⊢ (𝜑 → (𝑂 ∈ (InitO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})) |
6 | elrabi 3690 | . . . 4 ⊢ (𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)} → 𝑂 ∈ 𝐵) | |
7 | 5, 6 | biimtrdi 253 | . . 3 ⊢ (𝜑 → (𝑂 ∈ (InitO‘𝐶) → 𝑂 ∈ 𝐵)) |
8 | 7 | imp 406 | . 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → 𝑂 ∈ 𝐵) |
9 | 1 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝐶 ∈ Cat) |
10 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → 𝑂 ∈ 𝐵) | |
11 | 2, 3, 9, 10 | isinito 18050 | . . . 4 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
12 | 11 | biimpd 229 | . . 3 ⊢ ((𝜑 ∧ 𝑂 ∈ 𝐵) → (𝑂 ∈ (InitO‘𝐶) → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
13 | 12 | impancom 451 | . 2 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 → ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
14 | 8, 13 | jcai 516 | 1 ⊢ ((𝜑 ∧ 𝑂 ∈ (InitO‘𝐶)) → (𝑂 ∈ 𝐵 ∧ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑂𝐻𝑏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∃!weu 2566 ∀wral 3059 {crab 3433 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 Hom chom 17309 Catccat 17709 InitOcinito 18035 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-inito 18038 |
This theorem is referenced by: initoid 18055 initoo 18061 initoeu1 18065 initoeu2 18070 |
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