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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 18081 | . . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) |
| 5 | euex 2604 | . . 3 ⊢ (∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵) → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) | |
| 6 | 4, 5 | syl 18 | . 2 ⊢ (𝜑 → ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) |
| 7 | eqid 2762 | . . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
| 8 | eqid 2762 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
| 9 | termoo 18071 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (TermO‘𝐶) → 𝐴 ∈ (Base‘𝐶))) | |
| 10 | 1, 2, 9 | sylc 66 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶)) |
| 11 | termoo 18071 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (TermO‘𝐶) → 𝐵 ∈ (Base‘𝐶))) | |
| 12 | 1, 3, 11 | sylc 66 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶)) |
| 13 | 7, 8, 1, 10, 12 | cic 17862 | . 2 ⊢ (𝜑 → (𝐴( ≃𝑐 ‘𝐶)𝐵 ↔ ∃𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵))) |
| 14 | 6, 13 | mpbird 260 | 1 ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∃wex 1808 ∈ wcel 2142 ∃!weu 2595 class class class wbr 5108 ‘cfv 6536 (class class class)co 7412 Basecbs 17275 Catccat 17726 Isociso 17809 ≃𝑐 ccic 17858 TermOctermo 18045 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7984 df-2nd 7985 df-supp 8155 df-cat 17730 df-cid 17731 df-sect 17810 df-inv 17811 df-iso 17812 df-cic 17859 df-termo 18048 |
| This theorem is used by: nzerooringczr 21641 termcterm2 50320 termcciso 50322 |
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