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Theorem txswaphmeo 23725
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 481 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 simpr 483 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
31, 2cnmpt2nd 23589 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐾))
41, 2cnmpt1st 23588 . . 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 23593 . 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 5709 . . . . . . . . 9 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
76ancoms 457 . . . . . . . 8 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
87adantl 480 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ)) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
98ralrimivva 3191 . . . . . 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 2725 . . . . . . 7 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩)
1110fmpo 8068 . . . . . 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 5709 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1413ancoms 457 . . . . . . . 8 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1514adantl 480 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋)) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1615ralrimivva 3191 . . . . . 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 2725 . . . . . . 7 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)
1817fmpo 8068 . . . . . 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 23724 . . . . . 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 23724 . . . . . 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 7300 . . . . . 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 582 . . . 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 494 . . 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 23589 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ π‘₯) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
272, 1cnmpt1st 23588 . . . 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 23593 . . 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 2825 . 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 23679 . 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 581 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 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051  βŸ¨cop 4630   I cid 5569   Γ— cxp 5670  β—‘ccnv 5671   β†Ύ cres 5674   ∘ ccom 5676  βŸΆwf 6538  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7415   ∈ cmpo 7417  TopOnctopon 22828   Cn ccn 23144   Γ—t ctx 23480  Homeochmeo 23673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7418  df-oprab 7419  df-mpo 7420  df-1st 7989  df-2nd 7990  df-map 8843  df-topgen 17422  df-top 22812  df-topon 22829  df-bases 22865  df-cn 23147  df-tx 23482  df-hmeo 23675
This theorem is referenced by: (None)
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