Theorem eltx 23893

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 2760	. . . 4 ⊢ ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
2	1	txval 23889	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))))
3	2	eleq2d 2846	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)))))
4	1	txbasex 23891	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → ran (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5		eltg2b 23283	. . . 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 18	. . 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 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
8		vex 3454	. . . . . . 7 ⊢ 𝑦 ∈ V
9	7, 8	xpex 7768	. . . . . 6 ⊢ (𝑥 × 𝑦) ∈ V
10	9	rgen2w 3081	. . . . 5 ⊢ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝐾 (𝑥 × 𝑦) ∈ V
11		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝐽, 𝑦 ∈ 𝐾 ↦ (𝑥 × 𝑦))
12		eleq2 2849	. . . . . . 7 ⊢ (𝑧 = (𝑥 × 𝑦) → (𝑝 ∈ 𝑧 ↔ 𝑝 ∈ (𝑥 × 𝑦)))
13		sseq1 3933	. . . . . . 7 ⊢ (𝑧 = (𝑥 × 𝑦) → (𝑧 ⊆ 𝑆 ↔ (𝑥 × 𝑦) ⊆ 𝑆))
14	12, 13	anbi12d 644	. . . . . 6 ⊢ (𝑧 = (𝑥 × 𝑦) → ((𝑝 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑆) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
15	11, 14	rexrnmpo 7561	. . . . 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 3108	. . 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 290	. 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 282	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐾 ∈ 𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝 ∈ 𝑆 ∃𝑥 ∈ 𝐽 ∃𝑦 ∈ 𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895   × cxp 5648  ran crn 5651  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  topGenctg 17614   ×_t ctx 23885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-topgen 17620  df-tx 23887

This theorem is used by:  txcls  23929  txcnpi  23933  txdis  23957  txindis  23959  txdis1cn  23960  txlly  23961  txnlly  23962  txtube  23965  txcmplem1  23966  hausdiag  23970  tx1stc  23975  qustgplem  24446  txomap  34513  cvmlift2lem10  36164