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Mirrors > Home > MPE Home > Th. List > zerooval | Structured version Visualization version GIF version |
Description: The value of the zero object function, i.e. the set of all zero objects of a category. (Contributed by AV, 3-Apr-2020.) |
Ref | Expression |
---|---|
initoval.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
initoval.b | ⊢ 𝐵 = (Base‘𝐶) |
initoval.h | ⊢ 𝐻 = (Hom ‘𝐶) |
Ref | Expression |
---|---|
zerooval | ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-zeroo 17832 | . 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))) | |
2 | fveq2 6839 | . . 3 ⊢ (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶)) | |
3 | fveq2 6839 | . . 3 ⊢ (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶)) | |
4 | 2, 3 | ineq12d 4171 | . 2 ⊢ (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
5 | initoval.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
6 | fvex 6852 | . . . 4 ⊢ (InitO‘𝐶) ∈ V | |
7 | 6 | inex1 5272 | . . 3 ⊢ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V) |
9 | 1, 4, 5, 8 | fvmptd3 6968 | 1 ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∩ cin 3907 ‘cfv 6493 Basecbs 17043 Hom chom 17104 Catccat 17504 InitOcinito 17827 TermOctermo 17828 ZeroOczeroo 17829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-zeroo 17832 |
This theorem is referenced by: iszeroo 17844 iszeroi 17855 |
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