![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5600 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝑅 Se 𝐴) | |
2 | oicl.1 | . . . 4 ⊢ 𝐹 = OrdIso(𝑅, 𝐴) | |
3 | 2 | ordtype 9476 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
4 | 3 | ancoms 460 | . 2 ⊢ ((𝑅 Se 𝐴 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
5 | 1, 4 | sylan 581 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 We 𝐴) → 𝐹 Isom E , 𝑅 (dom 𝐹, 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 E cep 5540 Se wse 5590 We wwe 5591 dom cdm 5637 Isom wiso 6501 OrdIsocoi 9453 |
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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
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 2941 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-oi 9454 |
This theorem is referenced by: oien 9482 wofib 9489 cantnfle 9615 cantnflt 9616 cantnflt2 9617 cantnfp1lem3 9624 cantnflem1b 9630 cantnflem1d 9632 cantnflem1 9633 wemapwe 9641 cnfcomlem 9643 cnfcom 9644 cnfcom3lem 9647 infxpenlem 9957 finnisoeu 10057 dfac12lem2 10088 cofsmo 10213 fpwwe2lem5 10579 fpwwe2lem6 10580 fpwwe2lem8 10582 pwfseqlem5 10607 fz1isolem 14369 |
Copyright terms: Public domain | W3C validator |