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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 18003 | . . 3 ⊢ (𝜑 → ∃!𝑓 𝑓 ∈ (𝐴(Iso‘𝐶)𝐵)) |
5 | euex 2565 | . . 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 2725 | . . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶) | |
8 | eqid 2725 | . . 3 ⊢ (Base‘𝐶) = (Base‘𝐶) | |
9 | initoo 17999 | . . . 4 ⊢ (𝐶 ∈ Cat → (𝐴 ∈ (InitO‘𝐶) → 𝐴 ∈ (Base‘𝐶))) | |
10 | 1, 2, 9 | sylc 65 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Base‘𝐶)) |
11 | initoo 17999 | . . . 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 17785 | . 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 1773 ∈ wcel 2098 ∃!weu 2556 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 Catccat 17647 Isociso 17732 ≃𝑐 ccic 17781 InitOcinito 17973 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-supp 8166 df-cat 17651 df-cid 17652 df-sect 17733 df-inv 17734 df-iso 17735 df-cic 17782 df-inito 17976 |
This theorem is referenced by: nzerooringczr 21423 |
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