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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 17124 | . 2 ⊢ ZeroO = (𝑐 ∈ Cat ↦ ((InitO‘𝑐) ∩ (TermO‘𝑐))) | |
2 | fveq2 6497 | . . 3 ⊢ (𝑐 = 𝐶 → (InitO‘𝑐) = (InitO‘𝐶)) | |
3 | fveq2 6497 | . . 3 ⊢ (𝑐 = 𝐶 → (TermO‘𝑐) = (TermO‘𝐶)) | |
4 | 2, 3 | ineq12d 4072 | . 2 ⊢ (𝑐 = 𝐶 → ((InitO‘𝑐) ∩ (TermO‘𝑐)) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
5 | initoval.c | . 2 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
6 | fvex 6510 | . . . 4 ⊢ (InitO‘𝐶) ∈ V | |
7 | 6 | inex1 5075 | . . 3 ⊢ ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ((InitO‘𝐶) ∩ (TermO‘𝐶)) ∈ V) |
9 | 1, 4, 5, 8 | fvmptd3 6616 | 1 ⊢ (𝜑 → (ZeroO‘𝐶) = ((InitO‘𝐶) ∩ (TermO‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1508 ∈ wcel 2051 Vcvv 3410 ∩ cin 3823 ‘cfv 6186 Basecbs 16338 Hom chom 16431 Catccat 16806 InitOcinito 17119 TermOctermo 17120 ZeroOczeroo 17121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pr 5183 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ral 3088 df-rex 3089 df-rab 3092 df-v 3412 df-sbc 3677 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-iota 6150 df-fun 6188 df-fv 6194 df-zeroo 17124 |
This theorem is referenced by: iszeroo 17133 iszeroi 17140 |
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