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Mirrors > Home > MPE Home > Th. List > termoo | Structured version Visualization version GIF version |
Description: A terminal object is an object. (Contributed by AV, 18-Apr-2020.) |
Ref | Expression |
---|---|
termoo | ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2779 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
2 | eqid 2779 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
3 | id 22 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
4 | 1, 2, 3 | istermoi 17122 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂))) |
5 | 4 | simpld 487 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)) |
6 | 5 | ex 405 | 1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2050 ∃!weu 2583 ∀wral 3089 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 Hom chom 16432 Catccat 16793 TermOctermo 17107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3418 df-sbc 3683 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-ov 6979 df-termo 17110 |
This theorem is referenced by: 2termoinv 17135 termoeu1w 17137 |
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