Theorem txswaphmeo 23945

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 487	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐽 ∈ (TopOn‘𝑋))
2		simpr 489	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → 𝐾 ∈ (TopOn‘𝑌))
3	1, 2	cnmpt2nd 23809	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
4	1, 2	cnmpt1st 23808	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
5	1, 2, 3, 4	cnmpt2t 23813	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐽 ×t 𝐾) Cn (𝐾 ×t 𝐽)))
6		opelxpi 5702	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
7	6	ancoms 463	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
8	7	adantl 486	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌)) → ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
9	8	ralrimivva 3215	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋))
10		eqid 2770	. . . . . . 7 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)
11	10	fmpo 8068	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ⟨𝑦, 𝑥⟩ ∈ (𝑌 × 𝑋) ↔ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
12	9, 11	sylib 221	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩):(𝑋 × 𝑌)⟶(𝑌 × 𝑋))
13		opelxpi 5702	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
14	13	ancoms 463	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
15	14	adantl 486	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋)) → ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
16	15	ralrimivva 3215	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌))
17		eqid 2770	. . . . . . 7 ⊢ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)
18	17	fmpo 8068	. . . . . 6 ⊢ (∀𝑦 ∈ 𝑌 ∀𝑥 ∈ 𝑋 ⟨𝑥, 𝑦⟩ ∈ (𝑋 × 𝑌) ↔ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
19	16, 18	sylib 221	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩):(𝑌 × 𝑋)⟶(𝑋 × 𝑌))
20		txswaphmeolem 23944	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∘ (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩)) = ( I ↾ (𝑌 × 𝑋))
21		txswaphmeolem 23944	. . . . . 6 ⊢ ((𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∘ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩)) = ( I ↾ (𝑋 × 𝑌))
22		fcof1o 7298	. . . . . 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 717	. . . . 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 595	. . . 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 500	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) = (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩))
26	2, 1	cnmpt2nd 23809	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
27	2, 1	cnmpt1st 23808	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ 𝑦) ∈ ((𝐾 ×t 𝐽) Cn 𝐾))
28	2, 1, 26, 27	cnmpt2t 23813	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑦 ∈ 𝑌, 𝑥 ∈ 𝑋 ↦ ⟨𝑥, 𝑦⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
29	25, 28	eqeltrd 2870	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ⟨𝑦, 𝑥⟩) ∈ ((𝐾 ×t 𝐽) Cn (𝐽 ×t 𝐾)))
30		ishmeo 23899	. 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 594	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 400   = wceq 1568   ∈ wcel 2150  ∀wral 3086  ⟨cop 4600   I cid 5559   × cxp 5663  ^◡ccnv 5664   ↾ cres 5667   ∘ ccom 5669  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7414   ∈ cmpo 7416  TopOnctopon 23050   Cn ccn 23364   ×_t ctx 23700  Homeochmeo 23893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-map 8829  df-topgen 17499  df-top 23034  df-topon 23051  df-bases 23086  df-cn 23367  df-tx 23702  df-hmeo 23895

This theorem is referenced by: (None)