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Theorem txswaphmeo 21513
Description: There is a homeomorphism from 𝑋 × 𝑌 to 𝑌 × 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
txswaphmeo ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))
Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Proof of Theorem txswaphmeo
StepHypRef Expression
1 simpl 473 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
2 simpr 477 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ (TopOn‘𝑌))
31, 2cnmpt2nd 21377 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
41, 2cnmpt1st 21376 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
51, 2, 3, 4cnmpt2t 21381 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐾 ×t 𝐽)))
6 opelxpi 5113 . . . . . . . . 9 ((𝑦𝑌𝑥𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
76ancoms 469 . . . . . . . 8 ((𝑥𝑋𝑦𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
87adantl 482 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑥𝑋𝑦𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
98ralrimivva 2970 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑥𝑋𝑦𝑌𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
10 eqid 2626 . . . . . . 7 (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)
1110fmpt2 7183 . . . . . 6 (∀𝑥𝑋𝑦𝑌𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋) ↔ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
129, 11sylib 208 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
13 opelxpi 5113 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
1413ancoms 469 . . . . . . . 8 ((𝑦𝑌𝑥𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
1514adantl 482 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑦𝑌𝑥𝑋)) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
1615ralrimivva 2970 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑦𝑌𝑥𝑋𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
17 eqid 2626 . . . . . . 7 (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)
1817fmpt2 7183 . . . . . 6 (∀𝑦𝑌𝑥𝑋𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌) ↔ (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
1916, 18sylib 208 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
20 txswaphmeolem 21512 . . . . . 6 ((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∘ (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)) = ( I ↾ (𝑌 × 𝑋))
21 txswaphmeolem 21512 . . . . . 6 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
22 fcof1o 6506 . . . . . 6 ((((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋) ∧ (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌)) ∧ (((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∘ (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)) = ( I ↾ (𝑌 × 𝑋)) ∧ ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌)))) → ((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)–1-1-onto→(𝑌 × 𝑋) ∧ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)))
2320, 21, 22mpanr12 720 . . . . 5 (((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋) ∧ (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌)) → ((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)–1-1-onto→(𝑌 × 𝑋) ∧ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)))
2412, 19, 23syl2anc 692 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)–1-1-onto→(𝑌 × 𝑋) ∧ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)))
2524simprd 479 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩))
262, 1cnmpt2nd 21377 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋𝑥) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
272, 1cnmpt1st 21376 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋𝑦) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
282, 1, 26, 27cnmpt2t 21381 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
2925, 28eqeltrd 2704 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
30 ishmeo 21467 . 2 ((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)) ↔ ((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐾 ×t 𝐽)) ∧ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾))))
315, 29, 30sylanbrc 697 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  cop 4159   I cid 4989   × cxp 5077  ccnv 5078  cres 5081  ccom 5083  wf 5846  1-1-ontowf1o 5849  cfv 5850  (class class class)co 6605  cmpt2 6607  TopOnctopon 20613   Cn ccn 20933   ×t ctx 21268  Homeochmeo 21461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117  df-map 7805  df-topgen 16020  df-top 20616  df-bases 20617  df-topon 20618  df-cn 20936  df-tx 21270  df-hmeo 21463
This theorem is referenced by: (None)
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