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| 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 18039 | . . . . 5 ⊢ (𝜑 → (InitO‘𝐶) = {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)}) | 
| 5 | 4 | eleq2d 2826 | . . . 4 ⊢ (𝜑 → (𝑂 ∈ (InitO‘𝐶) ↔ 𝑂 ∈ {𝑎 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑎𝐻𝑏)})) | 
| 6 | elrabi 3686 | . . . 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 18042 | . . . 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 1539 ∈ wcel 2107 ∃!weu 2567 ∀wral 3060 {crab 3435 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Hom chom 17309 Catccat 17708 InitOcinito 18027 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-inito 18030 | 
| This theorem is referenced by: initoid 18047 initoo 18053 initoeu1 18057 initoeu2 18062 | 
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