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Mirrors > Home > MPE Home > Th. List > oiiso | Structured version Visualization version GIF version |
Description: The order isomorphism of the well-order 𝑅 on 𝐴 is an isomorphism. (Contributed by Mario Carneiro, 23-May-2015.) |
Ref | Expression |
---|---|
oicl.1 | ⊢ 𝐹 = OrdIso(𝑅, 𝐴) |
Ref | Expression |
---|---|
oiiso | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | exse 5638 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) | |
2 | oicl.1 | . . . 4 ⊢ 𝐹 = OrdIso(𝑅, 𝐴) | |
3 | 2 | ordtype 9523 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
4 | 3 | ancoms 459 | . 2 ⊢ ((𝑅 Se 𝐴 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
5 | 1, 4 | sylan 580 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 E cep 5578 Se wse 5628 We wwe 5629 dom cdm 5675 Isom wiso 6541 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-rep 5284 ax-sep 5298 ax-nul 5305 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-recs 8367 df-oi 9501 |
This theorem is referenced by: oien 9529 wofib 9536 cantnfle 9662 cantnflt 9663 cantnflt2 9664 cantnfp1lem3 9671 cantnflem1b 9677 cantnflem1d 9679 cantnflem1 9680 wemapwe 9688 cnfcomlem 9690 cnfcom 9691 cnfcom3lem 9694 infxpenlem 10004 finnisoeu 10104 dfac12lem2 10135 cofsmo 10260 fpwwe2lem5 10626 fpwwe2lem6 10627 fpwwe2lem8 10629 pwfseqlem5 10654 fz1isolem 14418 |
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