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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 17257 | . . 3 ⊢ (𝜑 → (InitO‘𝐶) = {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)}) |
5 | 4 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ 𝐼 ∈ {𝑖 ∈ 𝐵 ∣ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏)})) |
6 | isinito.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝐵) | |
7 | oveq1 7163 | . . . . . . 7 ⊢ (𝑖 = 𝐼 → (𝑖𝐻𝑏) = (𝐼𝐻𝑏)) | |
8 | 7 | eleq2d 2898 | . . . . . 6 ⊢ (𝑖 = 𝐼 → (ℎ ∈ (𝑖𝐻𝑏) ↔ ℎ ∈ (𝐼𝐻𝑏))) |
9 | 8 | eubidv 2672 | . . . . 5 ⊢ (𝑖 = 𝐼 → (∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
10 | 9 | ralbidv 3197 | . . . 4 ⊢ (𝑖 = 𝐼 → (∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝑖𝐻𝑏) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
11 | 10 | elrab3 3681 | . . 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 281 | 1 ⊢ (𝜑 → (𝐼 ∈ (InitO‘𝐶) ↔ ∀𝑏 ∈ 𝐵 ∃!ℎ ℎ ∈ (𝐼𝐻𝑏))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∃!weu 2653 ∀wral 3138 {crab 3142 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Hom chom 16576 Catccat 16935 InitOcinito 17248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-inito 17251 |
This theorem is referenced by: isinitoi 17263 initoeu2 17276 zrinitorngc 44291 irinitoringc 44360 zrninitoringc 44362 |
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