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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 6042 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | epse 5390 | . . 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 8790 | . . . . 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 577 | . . . 4 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴))) |
7 | ordsson 7320 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
8 | 4 | oismo 8799 | . . . . . 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 6897 | . . . . 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 496 | . . . 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 8788 | . . . . . 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 8774 | . . . . 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 1351 | . . . 4 ⊢ (Ord 𝐴 → dom OrdIso( E , 𝐴) = 𝐴) |
18 | isoeq4 6896 | . . . 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 6930 | . 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 1351 | 1 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ⊆ wss 3829 I cid 5311 E cep 5316 Se wse 5364 We wwe 5365 dom cdm 5407 ran crn 5408 ↾ cres 5409 Ord word 6028 Oncon0 6029 Isom wiso 6189 Smo wsmo 7786 OrdIsocoi 8768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-pss 3845 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-se 5367 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-isom 6197 df-riota 6937 df-wrecs 7750 df-smo 7787 df-recs 7812 df-oi 8769 |
This theorem is referenced by: hsmexlem5 9650 |
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