Theorem eltx 23506

Description: A set in a product is open iff each point is surrounded by an open rectangle. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Assertion

Ref	Expression
eltx	⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))

Distinct variable groups:   𝑥,𝑝,𝑦,𝐽   𝐾,𝑝,𝑥,𝑦   𝑆,𝑝,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑝)   𝑊(𝑥,𝑦,𝑝)

Proof of Theorem eltx

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2735	. . . 4 ⊢ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
2	1	txval 23502	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))))
3	2	eleq2d 2820	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)))))
4	1	txbasex 23504	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5		eltg2b 22897	. . . 4 ⊢ (ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) ∈ V → (𝑆 ∈ (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝 ∈ 𝑆 ∃𝑧 ∈ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆)))
6	4, 5	syl 17	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝 ∈ 𝑆 ∃𝑧 ∈ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆)))
7		vex 3463	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3463	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	xpex 7747	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
10	9	rgen2w 3056	. . . . 5 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐾 (𝑥 × 𝑦) ∈ V
11		eqid 2735	. . . . . 6 ⊢ (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
12		eleq2 2823	. . . . . . 7 ⊢ (𝑧 = (𝑥 × 𝑦) → (𝑝 ∈ 𝑧 ↔ 𝑝 ∈ (𝑥 × 𝑦)))
13		sseq1 3984	. . . . . . 7 ⊢ (𝑧 = (𝑥 × 𝑦) → (𝑧 ⊆ 𝑆 ↔ (𝑥 × 𝑦) ⊆ 𝑆))
14	12, 13	anbi12d 632	. . . . . 6 ⊢ (𝑧 = (𝑥 × 𝑦) → ((𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
15	11, 14	rexrnmpo 7547	. . . . 5 ⊢ (∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐾 (𝑥 × 𝑦) ∈ V → (∃𝑧 ∈ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
16	10, 15	ax-mp 5	. . . 4 ⊢ (∃𝑧 ∈ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) ↔ ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆))
17	16	ralbii 3082	. . 3 ⊢ (∀𝑝 ∈ 𝑆 ∃𝑧 ∈ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))(𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆))
18	6, 17	bitrdi 287	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
19	3, 18	bitrd 279	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ⊆ wss 3926   × cxp 5652  ran crn 5655  ‘cfv 6531  (class class class)co 7405   ∈ cmpo 7407  topGenctg 17451   ×_t ctx 23498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-topgen 17457  df-tx 23500

This theorem is referenced by:  txcls  23542  txcnpi  23546  txdis  23570  txindis  23572  txdis1cn  23573  txlly  23574  txnlly  23575  txtube  23578  txcmplem1  23579  hausdiag  23583  tx1stc  23588  qustgplem  24059  txomap  33865  cvmlift2lem10  35334