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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 6374 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | epse 5658 | . . 3 ⊢ E Se 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (Ord 𝐴 → E Se 𝐴) |
4 | eqid 2732 | . . . . . 6 ⊢ OrdIso( E , 𝐴) = OrdIso( E , 𝐴) | |
5 | 4 | oiiso2 9522 | . . . . 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 586 | . . . 4 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴))) |
7 | ordsson 7766 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
8 | 4 | oismo 9531 | . . . . . 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 7314 | . . . . 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 504 | . . . 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 231 | . . 3 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), 𝐴)) |
13 | 4 | oicl 9520 | . . . . . 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 9506 | . . . . 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 1371 | . . . 4 ⊢ (Ord 𝐴 → dom OrdIso( E , 𝐴) = 𝐴) |
18 | isoeq4 7313 | . . . 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 231 | . 2 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (𝐴, 𝐴)) |
21 | weniso 7347 | . 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 1371 | 1 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ⊆ wss 3947 I cid 5572 E cep 5578 Se wse 5628 We wwe 5629 dom cdm 5675 ran crn 5676 ↾ cres 5677 Ord word 6360 Oncon0 6361 Isom wiso 6541 Smo wsmo 8341 OrdIsocoi 9500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-smo 8342 df-recs 8367 df-oi 9501 |
This theorem is referenced by: hsmexlem5 10421 |
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