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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 17446 | . 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))) | |
2 | fveq2 6695 | . . 3 ⊢ (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶)) | |
3 | fveq2 6695 | . . 3 ⊢ (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶)) | |
4 | 2, 3 | ineq12d 4114 | . 2 ⊢ (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
5 | initoval.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
6 | fvex 6708 | . . . 4 ⊢ (InitO‘𝐶) ∈ V | |
7 | 6 | inex1 5195 | . . 3 ⊢ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V) |
9 | 1, 4, 5, 8 | fvmptd3 6819 | 1 ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∩ cin 3852 ‘cfv 6358 Basecbs 16666 Hom chom 16760 Catccat 17121 InitOcinito 17441 TermOctermo 17442 ZeroOczeroo 17443 |
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-zeroo 17446 |
This theorem is referenced by: iszeroo 17458 iszeroi 17469 |
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