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Theorem txswaphmeo 12479
 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 108 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
2 simpr 109 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ (TopOn‘𝑌))
31, 2cnmpt2nd 12447 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
41, 2cnmpt1st 12446 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
51, 2, 3, 4cnmpt2t 12451 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐾 ×t 𝐽)))
6 opelxpi 4566 . . . . . . . . 9 ((𝑦𝑌𝑥𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
76ancoms 266 . . . . . . . 8 ((𝑥𝑋𝑦𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
87adantl 275 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑥𝑋𝑦𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
98ralrimivva 2512 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑥𝑋𝑦𝑌𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
10 eqid 2137 . . . . . . 7 (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)
1110fmpo 6092 . . . . . 6 (∀𝑥𝑋𝑦𝑌𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋) ↔ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
129, 11sylib 121 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
13 opelxpi 4566 . . . . . . . . 9 ((𝑥𝑋𝑦𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
1413ancoms 266 . . . . . . . 8 ((𝑦𝑌𝑥𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
1514adantl 275 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑦𝑌𝑥𝑋)) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
1615ralrimivva 2512 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑦𝑌𝑥𝑋𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
17 eqid 2137 . . . . . . 7 (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)
1817fmpo 6092 . . . . . 6 (∀𝑦𝑌𝑥𝑋𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌) ↔ (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
1916, 18sylib 121 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
20 txswaphmeolem 12478 . . . . . 6 ((𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∘ (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩)) = ( I ↾ (𝑌 × 𝑋))
21 txswaphmeolem 12478 . . . . . 6 ((𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
22 fcof1o 5683 . . . . . 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 435 . . . . 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 408 . . . 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 113 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩))
262, 1cnmpt2nd 12447 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋𝑥) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
272, 1cnmpt1st 12446 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋𝑦) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
282, 1, 26, 27cnmpt2t 12451 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦𝑌, 𝑥𝑋 ↦ ⟨𝑥, 𝑦⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
2925, 28eqeltrd 2214 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
30 ishmeo 12462 . 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 413 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥𝑋, 𝑦𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ⟨cop 3525   I cid 4205   × cxp 4532  ◡ccnv 4533   ↾ cres 4536   ∘ ccom 4538  ⟶wf 5114  –1-1-onto→wf1o 5117  ‘cfv 5118  (class class class)co 5767   ∈ cmpo 5769  TopOnctopon 12166   Cn ccn 12343   ×t ctx 12410  Homeochmeo 12458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-map 6537  df-topgen 12130  df-top 12154  df-topon 12167  df-bases 12199  df-cn 12346  df-tx 12411  df-hmeo 12459 This theorem is referenced by: (None)
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