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Theorem txswaphmeo 23300
Description: There is a homeomorphism from 𝑋 Γ— π‘Œ to π‘Œ Γ— 𝑋. (Contributed by Mario Carneiro, 21-Mar-2015.)
Assertion
Ref Expression
txswaphmeo ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∈ ((𝐽 Γ—t 𝐾)Homeo(𝐾 Γ—t 𝐽)))
Distinct variable groups:   π‘₯,𝑦,𝐽   π‘₯,𝐾,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦
Proof of Theorem txswaphmeo
StepHypRef Expression
1 simpl 483 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 simpr 485 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
31, 2cnmpt2nd 23164 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐾))
41, 2cnmpt1st 23163 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
51, 2, 3, 4cnmpt2t 23168 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐾 Γ—t 𝐽)))
6 opelxpi 5712 . . . . . . . . 9 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
76ancoms 459 . . . . . . . 8 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
87adantl 482 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ)) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
98ralrimivva 3200 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
10 eqid 2732 . . . . . . 7 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩)
1110fmpo 8050 . . . . . 6 (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ π‘Œ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋) ↔ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩):(𝑋 Γ— π‘Œ)⟢(π‘Œ Γ— 𝑋))
129, 11sylib 217 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩):(𝑋 Γ— π‘Œ)⟢(π‘Œ Γ— 𝑋))
13 opelxpi 5712 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1413ancoms 459 . . . . . . . 8 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1514adantl 482 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋)) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1615ralrimivva 3200 . . . . . 6 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
17 eqid 2732 . . . . . . 7 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)
1817fmpo 8050 . . . . . 6 (βˆ€π‘¦ ∈ π‘Œ βˆ€π‘₯ ∈ 𝑋 ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ) ↔ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©):(π‘Œ Γ— 𝑋)⟢(𝑋 Γ— π‘Œ))
1916, 18sylib 217 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©):(π‘Œ Γ— 𝑋)⟢(𝑋 Γ— π‘Œ))
20 txswaphmeolem 23299 . . . . . 6 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∘ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)) = ( I β†Ύ (π‘Œ Γ— 𝑋))
21 txswaphmeolem 23299 . . . . . 6 ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©) ∘ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩)) = ( I β†Ύ (𝑋 Γ— π‘Œ))
22 fcof1o 7290 . . . . . 6 ((((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩):(𝑋 Γ— π‘Œ)⟢(π‘Œ Γ— 𝑋) ∧ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©):(π‘Œ Γ— 𝑋)⟢(𝑋 Γ— π‘Œ)) ∧ (((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∘ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)) = ( I β†Ύ (π‘Œ Γ— 𝑋)) ∧ ((𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©) ∘ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩)) = ( I β†Ύ (𝑋 Γ— π‘Œ)))) β†’ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩):(𝑋 Γ— π‘Œ)–1-1-ontoβ†’(π‘Œ Γ— 𝑋) ∧ β—‘(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)))
2320, 21, 22mpanr12 703 . . . . 5 (((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩):(𝑋 Γ— π‘Œ)⟢(π‘Œ Γ— 𝑋) ∧ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©):(π‘Œ Γ— 𝑋)⟢(𝑋 Γ— π‘Œ)) β†’ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩):(𝑋 Γ— π‘Œ)–1-1-ontoβ†’(π‘Œ Γ— 𝑋) ∧ β—‘(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)))
2412, 19, 23syl2anc 584 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩):(𝑋 Γ— π‘Œ)–1-1-ontoβ†’(π‘Œ Γ— 𝑋) ∧ β—‘(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)))
2524simprd 496 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ β—‘(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©))
262, 1cnmpt2nd 23164 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ π‘₯) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
272, 1cnmpt1st 23163 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ 𝑦) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐾))
282, 1, 26, 27cnmpt2t 23168 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©) ∈ ((𝐾 Γ—t 𝐽) Cn (𝐽 Γ—t 𝐾)))
2925, 28eqeltrd 2833 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ β—‘(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∈ ((𝐾 Γ—t 𝐽) Cn (𝐽 Γ—t 𝐾)))
30 ishmeo 23254 . 2 ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∈ ((𝐽 Γ—t 𝐾)Homeo(𝐾 Γ—t 𝐽)) ↔ ((π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∈ ((𝐽 Γ—t 𝐾) Cn (𝐾 Γ—t 𝐽)) ∧ β—‘(π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∈ ((𝐾 Γ—t 𝐽) Cn (𝐽 Γ—t 𝐾))))
315, 29, 30sylanbrc 583 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) ∈ ((𝐽 Γ—t 𝐾)Homeo(𝐾 Γ—t 𝐽)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βŸ¨cop 4633   I cid 5572   Γ— cxp 5673  β—‘ccnv 5674   β†Ύ cres 5677   ∘ ccom 5679  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  TopOnctopon 22403   Cn ccn 22719   Γ—t ctx 23055  Homeochmeo 23248
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-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  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-id 5573  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-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818  df-topgen 17385  df-top 22387  df-topon 22404  df-bases 22440  df-cn 22722  df-tx 23057  df-hmeo 23250
This theorem is referenced by: (None)
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