Theorem txswaphmeo 23156

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)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨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

Step	Hyp	Ref	Expression
1		simpl 483	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
2		simpr 485	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ (TopOn‘𝑌))
3	1, 2	cnmpt2nd 23020	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4	1, 2	cnmpt1st 23019	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
5	1, 2, 3, 4	cnmpt2t 23024	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐾 ×t 𝐽)))
6		opelxpi 5670	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
7	6	ancoms 459	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
8	7	adantl 482	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
9	8	ralrimivva 3197	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
10		eqid 2736	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
11	10	fmpo 8000	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
12	9, 11	sylib 217	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
13		opelxpi 5670	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
14	13	ancoms 459	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
15	14	adantl 482	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
16	15	ralrimivva 3197	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
17		eqid 2736	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)
18	17	fmpo 8000	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌) ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
19	16, 18	sylib 217	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
20		txswaphmeolem 23155	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∘ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)) = ( I ↾ (𝑌 × 𝑋))
21		txswaphmeolem 23155	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
22		fcof1o 7242	. . . . . 6 ⊢ ((((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋) ∧ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌)) ∧ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∘ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)) = ( I ↾ (𝑌 × 𝑋)) ∧ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌)))) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)–1-1-onto→(𝑌 × 𝑋) ∧ ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)))
23	20, 21, 22	mpanr12 703	. . . . 5 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋) ∧ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)–1-1-onto→(𝑌 × 𝑋) ∧ ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)))
24	12, 19, 23	syl2anc 584	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)–1-1-onto→(𝑌 × 𝑋) ∧ ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)))
25	24	simprd 496	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩))
26	2, 1	cnmpt2nd 23020	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
27	2, 1	cnmpt1st 23019	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝑦) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
28	2, 1, 26, 27	cnmpt2t 23024	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
29	25, 28	eqeltrd 2838	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
30		ishmeo 23110	. 2 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)) ↔ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐾 ×t 𝐽)) ∧ ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾))))
31	5, 29, 30	sylanbrc 583	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾)Homeo(𝐾 ×t 𝐽)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ⟨cop 4592   I cid 5530   × cxp 5631  ^◡ccnv 5632   ↾ cres 5635   ∘ ccom 5637  ⟶wf 6492  –1-1-onto→wf1o 6495  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  TopOnctopon 22259   Cn ccn 22575   ×_t ctx 22911  Homeochmeo 23104

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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-cn 22578  df-tx 22913  df-hmeo 23106

This theorem is referenced by: (None)