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Mirrors > Home > MPE Home > Th. List > oiid | Structured version Visualization version GIF version |
Description: The order type of an ordinal under the ∈ order is itself, and the order isomorphism is the identity function. (Contributed by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
oiid | ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordwe 5989 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | epse 5338 | . . 3 ⊢ E Se 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (Ord 𝐴 → E Se 𝐴) |
4 | eqid 2778 | . . . . . 6 ⊢ OrdIso( E , 𝐴) = OrdIso( E , 𝐴) | |
5 | 4 | oiiso2 8725 | . . . . 5 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴))) |
6 | 1, 2, 5 | sylancl 580 | . . . 4 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴))) |
7 | ordsson 7267 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
8 | 4 | oismo 8734 | . . . . . 6 ⊢ (𝐴 ⊆ On → (Smo OrdIso( E , 𝐴) ∧ ran OrdIso( E , 𝐴) = 𝐴)) |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (Ord 𝐴 → (Smo OrdIso( E , 𝐴) ∧ ran OrdIso( E , 𝐴) = 𝐴)) |
10 | isoeq5 6843 | . . . . 5 ⊢ (ran OrdIso( E , 𝐴) = 𝐴 → (OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴)) ↔ OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴))) | |
11 | 9, 10 | simpl2im 499 | . . . 4 ⊢ (Ord 𝐴 → (OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴)) ↔ OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴))) |
12 | 6, 11 | mpbid 224 | . . 3 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴)) |
13 | 4 | oicl 8723 | . . . . . 6 ⊢ Ord dom OrdIso( E , 𝐴) |
14 | 13 | a1i 11 | . . . . 5 ⊢ (Ord 𝐴 → Ord dom OrdIso( E , 𝐴)) |
15 | id 22 | . . . . 5 ⊢ (Ord 𝐴 → Ord 𝐴) | |
16 | ordiso2 8709 | . . . . 5 ⊢ ((OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴) ∧ Ord dom OrdIso( E , 𝐴) ∧ Ord 𝐴) → dom OrdIso( E , 𝐴) = 𝐴) | |
17 | 12, 14, 15, 16 | syl3anc 1439 | . . . 4 ⊢ (Ord 𝐴 → dom OrdIso( E , 𝐴) = 𝐴) |
18 | isoeq4 6842 | . . . 4 ⊢ (dom OrdIso( E , 𝐴) = 𝐴 → (OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴) ↔ OrdIso( E , 𝐴) Isom E , E (𝐴, 𝐴))) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (Ord 𝐴 → (OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴) ↔ OrdIso( E , 𝐴) Isom E , E (𝐴, 𝐴))) |
20 | 12, 19 | mpbid 224 | . 2 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (𝐴, 𝐴)) |
21 | weniso 6876 | . 2 ⊢ (( E We 𝐴 ∧ E Se 𝐴 ∧ OrdIso( E , 𝐴) Isom E , E (𝐴, 𝐴)) → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) | |
22 | 1, 3, 20, 21 | syl3anc 1439 | 1 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ⊆ wss 3792 I cid 5260 E cep 5265 Se wse 5312 We wwe 5313 dom cdm 5355 ran crn 5356 ↾ cres 5357 Ord word 5975 Oncon0 5976 Isom wiso 6136 Smo wsmo 7725 OrdIsocoi 8703 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-wrecs 7689 df-smo 7726 df-recs 7751 df-oi 8704 |
This theorem is referenced by: hsmexlem5 9587 |
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