Theorem eltx 23576

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 2737	. . . 4 ⊢ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
2	1	txval 23572	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))))
3	2	eleq2d 2827	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)))))
4	1	txbasex 23574	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5		eltg2b 22966	. . . 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 3484	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3484	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	xpex 7773	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
10	9	rgen2w 3066	. . . . 5 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐾 (𝑥 × 𝑦) ∈ V
11		eqid 2737	. . . . . 6 ⊢ (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
12		eleq2 2830	. . . . . . 7 ⊢ (𝑧 = (𝑥 × 𝑦) → (𝑝 ∈ 𝑧 ↔ 𝑝 ∈ (𝑥 × 𝑦)))
13		sseq1 4009	. . . . . . 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 7573	. . . . 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 3093	. . 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 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951   × cxp 5683  ran crn 5686  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  topGenctg 17482   ×_t ctx 23568

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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-topgen 17488  df-tx 23570

This theorem is referenced by:  txcls  23612  txcnpi  23616  txdis  23640  txindis  23642  txdis1cn  23643  txlly  23644  txnlly  23645  txtube  23648  txcmplem1  23649  hausdiag  23653  tx1stc  23658  qustgplem  24129  txomap  33833  cvmlift2lem10  35317