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Mirrors > Home > MPE Home > Th. List > initoeu1w | Structured version Visualization version GIF version |
Description: Initial objects are essentially unique (weak form), i.e. if A and B are initial objects, then A and B are isomorphic. Proposition 7.3 (1) of [Adamek] p. 102. (Contributed by AV, 6-Apr-2020.) |
Ref | Expression |
---|---|
initoeu1.c | ⊢ (𝜑 → 𝐶 ∈ Cat) |
initoeu1.a | ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶)) |
initoeu1.b | ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶)) |
Ref | Expression |
---|---|
initoeu1w | ⊢ (𝜑 → 𝐴( ≃𝑐 ‘𝐶)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | initoeu1.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat) | |
2 | initoeu1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (InitO‘𝐶)) | |
3 | initoeu1.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (InitO‘𝐶)) | |
4 | 1, 2, 3 | initoeu1 17271 | . . 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 2821 | . . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
8 | eqid 2821 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | initoo 17267 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶))) | |
10 | 1, 2, 9 | sylc 65 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶)) |
11 | initoo 17267 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐵 ∈ (InitO‘𝐶) → 𝐵 ∈ (Base‘𝐶))) | |
12 | 1, 3, 11 | sylc 65 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (Base‘𝐶)) |
13 | 7, 8, 1, 10, 12 | cic 17069 | . 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 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Catccat 16935 Isociso 17016 ≃𝑐 ccic 17065 InitOcinito 17248 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-supp 7831 df-cat 16939 df-cid 16940 df-sect 17017 df-inv 17018 df-iso 17019 df-cic 17066 df-inito 17251 |
This theorem is referenced by: nzerooringczr 44363 |
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