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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 17733 | . . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) |
5 | euex 2577 | . . 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 2738 | . . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
8 | eqid 2738 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | termoo 17723 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶))) | |
10 | 1, 2, 9 | sylc 65 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶)) |
11 | termoo 17723 | . . . 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 17511 | . 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))) |
14 | 6, 13 | mpbird 256 | 1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wex 1782 ∈ wcel 2106 ∃!weu 2568 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Catccat 17373 Isociso 17458 ≃𝑐 ccic 17507 TermOctermo 17697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-supp 7978 df-cat 17377 df-cid 17378 df-sect 17459 df-inv 17460 df-iso 17461 df-cic 17508 df-termo 17700 |
This theorem is referenced by: nzerooringczr 45630 |
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