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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 2736 | . . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 2 | eqid 2736 | . . . 4 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶) | |
| 3 | id 22 | . . . 4 ⊢ (𝐶 ∈ Cat → 𝐶 ∈ Cat) | |
| 4 | 1, 2, 3 | istermoi 18046 | . . 3 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → (𝑂 ∈ (Base‘𝐶) ∧ ∀𝑏 ∈ (Base‘𝐶)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐶)𝑂))) | 
| 5 | 4 | simpld 494 | . 2 ⊢ ((𝐶 ∈ Cat ∧ 𝑂 ∈ (TermO‘𝐶)) → 𝑂 ∈ (Base‘𝐶)) | 
| 6 | 5 | ex 412 | 1 ⊢ (𝐶 ∈ Cat → (𝑂 ∈ (TermO‘𝐶) → 𝑂 ∈ (Base‘𝐶))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2107 ∃!weu 2567 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 Hom chom 17309 Catccat 17708 TermOctermo 18028 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fv 6568 df-ov 7435 df-termo 18031 | 
| This theorem is referenced by: 2termoinv 18063 termoeu1w 18065 | 
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