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Mirrors > Home > MPE Home > Th. List > iszeroo | Structured version Visualization version GIF version |
Description: The predicate "is a zero 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 |
---|---|
iszeroo | ⊢ (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶)))) |
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 | zerooval 17259 | . . 3 ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
5 | 4 | eleq2d 2898 | . 2 ⊢ (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ 𝐼 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)))) |
6 | elin 4169 | . 2 ⊢ (𝐼 ∈ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶))) | |
7 | 5, 6 | syl6bb 289 | 1 ⊢ (𝜑 → (𝐼 ∈ (ZeroO‘𝐶) ↔ (𝐼 ∈ (InitO‘𝐶) ∧ 𝐼 ∈ (TermO‘𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3935 ‘cfv 6355 Basecbs 16483 Hom chom 16576 Catccat 16935 InitOcinito 17248 TermOctermo 17249 ZeroOczeroo 17250 |
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-zeroo 17253 |
This theorem is referenced by: iszeroi 17269 zrzeroorngc 44293 |
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