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Theorem eltx 22168
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 2819 . . . 4 ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
21txval 22164 . . 3 ((𝐽𝑉𝐾𝑊) → (𝐽 ×t 𝐾) = (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))))
32eleq2d 2896 . 2 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ 𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)))))
41txbasex 22166 . . . 4 ((𝐽𝑉𝐾𝑊) → ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) ∈ V)
5 eltg2b 21559 . . . 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 3496 . . . . . . 7 𝑥 ∈ V
8 vex 3496 . . . . . . 7 𝑦 ∈ V
97, 8xpex 7468 . . . . . 6 (𝑥 × 𝑦) ∈ V
109rgen2w 3149 . . . . 5 𝑥𝐽𝑦𝐾 (𝑥 × 𝑦) ∈ V
11 eqid 2819 . . . . . 6 (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦)) = (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))
12 eleq2 2899 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑝𝑧𝑝 ∈ (𝑥 × 𝑦)))
13 sseq1 3990 . . . . . . 7 (𝑧 = (𝑥 × 𝑦) → (𝑧𝑆 ↔ (𝑥 × 𝑦) ⊆ 𝑆))
1412, 13anbi12d 632 . . . . . 6 (𝑧 = (𝑥 × 𝑦) → ((𝑝𝑧𝑧𝑆) ↔ (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
1511, 14rexrnmpo 7282 . . . . 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 3163 . . 3 (∀𝑝𝑆𝑧 ∈ ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))(𝑝𝑧𝑧𝑆) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆))
186, 17syl6bb 289 . 2 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (topGen‘ran (𝑥𝐽, 𝑦𝐾 ↦ (𝑥 × 𝑦))) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
193, 18bitrd 281 1 ((𝐽𝑉𝐾𝑊) → (𝑆 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑝𝑆𝑥𝐽𝑦𝐾 (𝑝 ∈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ⊆ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wcel 2107  wral 3136  wrex 3137  Vcvv 3493  wss 3934   × cxp 5546  ran crn 5549  cfv 6348  (class class class)co 7148  cmpo 7150  topGenctg 16703   ×t ctx 22160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-topgen 16709  df-tx 22162
This theorem is referenced by:  txcls  22204  txcnpi  22208  txdis  22232  txindis  22234  txdis1cn  22235  txlly  22236  txnlly  22237  txtube  22240  txcmplem1  22241  hausdiag  22245  tx1stc  22250  qustgplem  22721  txomap  31091  cvmlift2lem10  32552
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