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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 5639 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) | |
2 | oicl.1 | . . . 4 ⊢ 𝐹 = OrdIso(𝑅, 𝐴) | |
3 | 2 | ordtype 9526 | . . 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 5579 Se wse 5629 We wwe 5630 dom cdm 5676 Isom wiso 6544 OrdIsocoi 9503 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7364 df-ov 7411 df-2nd 7975 df-frecs 8265 df-wrecs 8296 df-recs 8370 df-oi 9504 |
This theorem is referenced by: oien 9532 wofib 9539 cantnfle 9665 cantnflt 9666 cantnflt2 9667 cantnfp1lem3 9674 cantnflem1b 9680 cantnflem1d 9682 cantnflem1 9683 wemapwe 9691 cnfcomlem 9693 cnfcom 9694 cnfcom3lem 9697 infxpenlem 10007 finnisoeu 10107 dfac12lem2 10138 cofsmo 10263 fpwwe2lem5 10629 fpwwe2lem6 10630 fpwwe2lem8 10632 pwfseqlem5 10657 fz1isolem 14421 |
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