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Theorem eltx 21780
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 2778 . . . 4 ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
21txval 21776 . . 3 ((𝐽𝑉𝐾𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))))
32eleq2d 2845 . 2 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)))))
41txbasex 21778 . . . 4 ((𝐽𝑉𝐾𝑊) → ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5 eltg2b 21171 . . . 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 3401 . . . . . . 7 𝑥 ∈ V
8 vex 3401 . . . . . . 7 𝑦 ∈ V
97, 8xpex 7240 . . . . . 6 (𝑥 × 𝑦) ∈ V
109rgen2w 3107 . . . . 5 𝑥𝐽𝑦𝐾 (𝑥 × 𝑦) ∈ V
11 eqid 2778 . . . . . 6 (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
12 eleq2 2848 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑝𝑧𝑝 ∈ (𝑥 × 𝑦)))
13 sseq1 3845 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑧𝑆 ↔ (𝑥 × 𝑦) ⊆ 𝑆))
1412, 13anbi12d 624 . . . . . 6 (𝑧 = (𝑥 × 𝑦) → ((𝑝𝑧𝑧𝑆) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
1511, 14rexrnmpt2 7053 . . . . 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 3162 . . 3 (∀𝑝𝑆𝑧 ∈ ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))(𝑝𝑧𝑧𝑆) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆))
186, 17syl6bb 279 . 2 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
193, 18bitrd 271 1 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wcel 2107  wral 3090  wrex 3091  Vcvv 3398  wss 3792   × cxp 5353  ran crn 5356  cfv 6135  (class class class)co 6922  cmpt2 6924  topGenctg 16484   ×t ctx 21772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446  df-topgen 16490  df-tx 21774
This theorem is referenced by:  txcls  21816  txcnpi  21820  txdis  21844  txindis  21846  txdis1cn  21847  txlly  21848  txnlly  21849  txtube  21852  txcmplem1  21853  hausdiag  21857  tx1stc  21862  qustgplem  22332  txomap  30499  cvmlift2lem10  31893
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