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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 6198 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | epse 5532 | . . 3 ⊢ E Se 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (Ord 𝐴 → E Se 𝐴) |
4 | eqid 2821 | . . . . . 6 ⊢ OrdIso( E , 𝐴) = OrdIso( E , 𝐴) | |
5 | 4 | oiiso2 8989 | . . . . 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 588 | . . . 4 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴))) |
7 | ordsson 7498 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
8 | 4 | oismo 8998 | . . . . . 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 7068 | . . . . 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 506 | . . . 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 234 | . . 3 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴)) |
13 | 4 | oicl 8987 | . . . . . 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 8973 | . . . . 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 1367 | . . . 4 ⊢ (Ord 𝐴 → dom OrdIso( E , 𝐴) = 𝐴) |
18 | isoeq4 7067 | . . . 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 234 | . 2 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (𝐴, 𝐴)) |
21 | weniso 7101 | . 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 1367 | 1 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ⊆ wss 3935 I cid 5453 E cep 5458 Se wse 5506 We wwe 5507 dom cdm 5549 ran crn 5550 ↾ cres 5551 Ord word 6184 Oncon0 6185 Isom wiso 6350 Smo wsmo 7976 OrdIsocoi 8967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-wrecs 7941 df-smo 7977 df-recs 8002 df-oi 8968 |
This theorem is referenced by: hsmexlem5 9846 |
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