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Theorem txswaphmeo 23664
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 482 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
2 simpr 484 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
31, 2cnmpt2nd 23528 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ 𝑦) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐾))
41, 2cnmpt1st 23527 . . 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 23532 . 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 5706 . . . . . . . . 9 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
76ancoms 458 . . . . . . . 8 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
87adantl 481 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ)) β†’ βŸ¨π‘¦, π‘₯⟩ ∈ (π‘Œ Γ— 𝑋))
98ralrimivva 3194 . . . . . 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 2726 . . . . . . 7 (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ βŸ¨π‘¦, π‘₯⟩)
1110fmpo 8053 . . . . . 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 5706 . . . . . . . . 9 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1413ancoms 458 . . . . . . . 8 ((𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1514adantl 481 . . . . . . 7 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) ∧ (𝑦 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋)) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝑋 Γ— π‘Œ))
1615ralrimivva 3194 . . . . . 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 2726 . . . . . . 7 (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©) = (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ ⟨π‘₯, π‘¦βŸ©)
1817fmpo 8053 . . . . . 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 23663 . . . . . 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 23663 . . . . . 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 702 . . . . 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 583 . . . 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 495 . . 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 23528 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝑦 ∈ π‘Œ, π‘₯ ∈ 𝑋 ↦ π‘₯) ∈ ((𝐾 Γ—t 𝐽) Cn 𝐽))
272, 1cnmpt1st 23527 . . . 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 23532 . . 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 2827 . 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 23618 . 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 582 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βŸ¨cop 4629   I cid 5566   Γ— cxp 5667  β—‘ccnv 5668   β†Ύ cres 5671   ∘ ccom 5673  βŸΆwf 6533  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  TopOnctopon 22767   Cn ccn 23083   Γ—t ctx 23419  Homeochmeo 23612
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824  df-topgen 17398  df-top 22751  df-topon 22768  df-bases 22804  df-cn 23086  df-tx 23421  df-hmeo 23614
This theorem is referenced by: (None)
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