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Theorem eltx 21742
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 2825 . . . 4 ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
21txval 21738 . . 3 ((𝐽𝑉𝐾𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))))
32eleq2d 2892 . 2 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)))))
41txbasex 21740 . . . 4 ((𝐽𝑉𝐾𝑊) → ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5 eltg2b 21134 . . . 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 3417 . . . . . . 7 𝑥 ∈ V
8 vex 3417 . . . . . . 7 𝑦 ∈ V
97, 8xpex 7223 . . . . . 6 (𝑥 × 𝑦) ∈ V
109rgen2w 3134 . . . . 5 𝑥𝐽𝑦𝐾 (𝑥 × 𝑦) ∈ V
11 eqid 2825 . . . . . 6 (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
12 eleq2 2895 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑝𝑧𝑝 ∈ (𝑥 × 𝑦)))
13 sseq1 3851 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑧𝑆 ↔ (𝑥 × 𝑦) ⊆ 𝑆))
1412, 13anbi12d 626 . . . . . 6 (𝑧 = (𝑥 × 𝑦) → ((𝑝𝑧𝑧𝑆) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
1511, 14rexrnmpt2 7036 . . . . 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 3189 . . 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 1658  wcel 2166  wral 3117  wrex 3118  Vcvv 3414  wss 3798   × cxp 5340  ran crn 5343  cfv 6123  (class class class)co 6905  cmpt2 6907  topGenctg 16451   ×t ctx 21734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-topgen 16457  df-tx 21736
This theorem is referenced by:  txcls  21778  txcnpi  21782  txdis  21806  txindis  21808  txdis1cn  21809  txlly  21810  txnlly  21811  txtube  21814  txcmplem1  21815  hausdiag  21819  tx1stc  21824  qustgplem  22294  txomap  30446  cvmlift2lem10  31840
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