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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 17978 | . 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))) | |
2 | fveq2 6896 | . . 3 ⊢ (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶)) | |
3 | fveq2 6896 | . . 3 ⊢ (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶)) | |
4 | 2, 3 | ineq12d 4211 | . 2 ⊢ (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
5 | initoval.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
6 | fvex 6909 | . . . 4 ⊢ (InitO‘𝐶) ∈ V | |
7 | 6 | inex1 5318 | . . 3 ⊢ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V) |
9 | 1, 4, 5, 8 | fvmptd3 7027 | 1 ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3461 ∩ cin 3943 ‘cfv 6549 Basecbs 17183 Hom chom 17247 Catccat 17647 InitOcinito 17973 TermOctermo 17974 ZeroOczeroo 17975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-zeroo 17978 |
This theorem is referenced by: iszeroo 17990 iszeroi 18001 |
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