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Mirrors > Home > MPE Home > Th. List > termoeu1w | Structured version Visualization version GIF version |
Description: Terminal objects are essentially unique (weak form), i.e. if A and B are terminal objects, then A and B are isomorphic. Proposition 7.6 of [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.) |
Ref | Expression |
---|---|
termoeu1.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
termoeu1.a | ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶)) |
termoeu1.b | ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶)) |
Ref | Expression |
---|---|
termoeu1w | ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | termoeu1.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | termoeu1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (TermO‘𝐶)) | |
3 | termoeu1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (TermO‘𝐶)) | |
4 | 1, 2, 3 | termoeu1 17280 | . . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) |
5 | euex 2662 | . . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) |
7 | eqid 2823 | . . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
8 | eqid 2823 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | termoo 17270 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶))) | |
10 | 1, 2, 9 | sylc 65 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶)) |
11 | termoo 17270 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶))) | |
12 | 1, 3, 11 | sylc 65 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶)) |
13 | 7, 8, 1, 10, 12 | cic 17071 | . 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))) |
14 | 6, 13 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1780 ∈ wcel 2114 ∃!weu 2653 class class class wbr 5068 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 Catccat 16937 Isociso 17018 ≃𝑐 ccic 17067 TermOctermo 17251 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-supp 7833 df-cat 16941 df-cid 16942 df-sect 17019 df-inv 17020 df-iso 17021 df-cic 17068 df-termo 17254 |
This theorem is referenced by: nzerooringczr 44350 |
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