MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eltx Structured version   Visualization version   GIF version

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 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (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.
StepHypRef Expression
1 eqid 2737 . . . 4 ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
21txval 23572 . . 3 ((𝐽𝑉𝐾𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))))
32eleq2d 2827 . 2 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)))))
41txbasex 23574 . . . 4 ((𝐽𝑉𝐾𝑊) → ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5 eltg2b 22966 . . . 4 (ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) ∈ V → (𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝𝑆𝑧 ∈ ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))(𝑝𝑧𝑧𝑆)))
64, 5syl 17 . . 3 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝𝑆𝑧 ∈ ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))(𝑝𝑧𝑧𝑆)))
7 vex 3484 . . . . . . 7 𝑥 ∈ V
8 vex 3484 . . . . . . 7 𝑦 ∈ V
97, 8xpex 7773 . . . . . 6 (𝑥 × 𝑦) ∈ V
109rgen2w 3066 . . . . 5 𝑥𝐽𝑦𝐾 (𝑥 × 𝑦) ∈ V
11 eqid 2737 . . . . . 6 (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
12 eleq2 2830 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑝𝑧𝑝 ∈ (𝑥 × 𝑦)))
13 sseq1 4009 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑧𝑆 ↔ (𝑥 × 𝑦) ⊆ 𝑆))
1412, 13anbi12d 632 . . . . . 6 (𝑧 = (𝑥 × 𝑦) → ((𝑝𝑧𝑧𝑆) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
1511, 14rexrnmpo 7573 . . . . 5 (∀𝑥𝐽𝑦𝐾 (𝑥 × 𝑦) ∈ V → (∃𝑧 ∈ ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))(𝑝𝑧𝑧𝑆) ↔ ∃𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
1610, 15ax-mp 5 . . . 4 (∃𝑧 ∈ ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))(𝑝𝑧𝑧𝑆) ↔ ∃𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆))
1716ralbii 3093 . . 3 (∀𝑝𝑆𝑧 ∈ ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))(𝑝𝑧𝑧𝑆) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆))
186, 17bitrdi 287 . 2 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
193, 18bitrd 279 1 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator