Theorem eltx 15012

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 2230	. . . 4 ⊢ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
2	1	txval 15008	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))))
3	2	eleq2d 2300	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)))))
4	1	txbasex 15010	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5		eltg2b 14807	. . . 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 14	. . 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 2804	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 2804	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	xpex 4844	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
10	9	rgen2w 2587	. . . . 5 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐾 (𝑥 × 𝑦) ∈ V
11		eqid 2230	. . . . . 6 ⊢ (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
12		eleq2 2294	. . . . . . 7 ⊢ (𝑧 = (𝑥 × 𝑦) → (𝑝 ∈ 𝑧 ↔ 𝑝 ∈ (𝑥 × 𝑦)))
13		sseq1 3249	. . . . . . 7 ⊢ (𝑧 = (𝑥 × 𝑦) → (𝑧 ⊆ 𝑆 ↔ (𝑥 × 𝑦) ⊆ 𝑆))
14	12, 13	anbi12d 473	. . . . . 6 ⊢ (𝑧 = (𝑥 × 𝑦) → ((𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
15	11, 14	rexrnmpo 6142	. . . . 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 2537	. . 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 196	. 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 188	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   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2201  ∀wral 2509  ∃wrex 2510  Vcvv 2801   ⊆ wss 3199   × cxp 4725  ran crn 4728  ‘cfv 5328  (class class class)co 6023   ∈ cmpo 6025  topGenctg 13360   ×_t ctx 15005

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-topgen 13366  df-tx 15006

This theorem is referenced by:  txdis  15030  txdis1cn  15031