Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isinito | Structured version Visualization version GIF version |
Description: The predicate "is an initial object" of a category. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
isinito.b | ⊢ 𝐵 = (Base‘𝐶) |
isinito.h | ⊢ 𝐻 = (Hom ‘𝐶) |
isinito.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
isinito.i | ⊢ (𝜑 → 𝐼 ∈ 𝐵) |
Ref | Expression |
---|---|
isinito | ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isinito.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | isinito.b | . . . 4 ⊢ 𝐵 = (Base‘𝐶) | |
3 | isinito.h | . . . 4 ⊢ 𝐻 = (Hom ‘𝐶) | |
4 | 1, 2, 3 | initoval 17453 | . . 3 ⊢ (𝜑 → (InitO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)}) |
5 | 4 | eleq2d 2816 | . 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)})) |
6 | isinito.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
7 | oveq1 7198 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏)) | |
8 | 7 | eleq2d 2816 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑖𝐻𝑏) ↔ ℎ ∈ (𝐼𝐻𝑏))) |
9 | 8 | eubidv 2585 | . . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
10 | 9 | ralbidv 3108 | . . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
11 | 10 | elrab3 3592 | . . 3 ⊢ (𝐼 ∈ 𝐵 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
12 | 6, 11 | syl 17 | . 2 ⊢ (𝜑 → (𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)} ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
13 | 5, 12 | bitrd 282 | 1 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∃!weu 2567 ∀wral 3051 {crab 3055 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 Hom chom 16760 Catccat 17121 InitOcinito 17441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6316 df-fun 6360 df-fv 6366 df-ov 7194 df-inito 17444 |
This theorem is referenced by: isinitoi 17459 initoeu2 17476 zrinitorngc 45174 irinitoringc 45243 zrninitoringc 45245 |
Copyright terms: Public domain | W3C validator |