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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 6369 | . 2 ⊢ (Ord 𝐴 → E We 𝐴) | |
2 | epse 5655 | . . 3 ⊢ E Se 𝐴 | |
3 | 2 | a1i 11 | . 2 ⊢ (Ord 𝐴 → E Se 𝐴) |
4 | eqid 2733 | . . . . . 6 ⊢ OrdIso( E , 𝐴) = OrdIso( E , 𝐴) | |
5 | 4 | oiiso2 9513 | . . . . 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 587 | . . . 4 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) Isom E , E (dom OrdIso( E , 𝐴), ran OrdIso( E , 𝐴))) |
7 | ordsson 7757 | . . . . . 6 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
8 | 4 | oismo 9522 | . . . . . 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 7305 | . . . . 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 505 | . . . 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 9511 | . . . . . 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 9497 | . . . . 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 1372 | . . . 4 ⊢ (Ord 𝐴 → dom OrdIso( E , 𝐴) = 𝐴) |
18 | isoeq4 7304 | . . . 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 7338 | . 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 1372 | 1 ⊢ (Ord 𝐴 → OrdIso( E , 𝐴) = ( I ↾ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ⊆ wss 3946 I cid 5569 E cep 5575 Se wse 5625 We wwe 5626 dom cdm 5672 ran crn 5673 ↾ cres 5674 Ord word 6355 Oncon0 6356 Isom wiso 6536 Smo wsmo 8332 OrdIsocoi 9491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7352 df-ov 7399 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-smo 8333 df-recs 8358 df-oi 9492 |
This theorem is referenced by: hsmexlem5 10412 |
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