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Theorem txrest 23655
Description: The subspace of a topological product space induced by a subset with a Cartesian product representation is a topological product of the subspaces induced by the subspaces of the terms of the products. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
txrest (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅t 𝐴) ×t (𝑆t 𝐵)))

Proof of Theorem txrest
Dummy variables 𝑠 𝑟 𝑢 𝑣 𝑥 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2735 . . . . . 6 ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
21txval 23588 . . . . 5 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
32adantr 480 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
43oveq1d 7446 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
51txbasex 23590 . . . 4 ((𝑅𝑉𝑆𝑊) → ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6 xpexg 7769 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ∈ V)
7 tgrest 23183 . . . 4 ((ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ∈ V ∧ (𝐴 × 𝐵) ∈ V) → (topGen‘(ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
85, 6, 7syl2an 596 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (topGen‘(ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
9 elrest 17474 . . . . . . . 8 ((ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ∈ V ∧ (𝐴 × 𝐵) ∈ V) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
105, 6, 9syl2an 596 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
11 vex 3482 . . . . . . . . . . 11 𝑟 ∈ V
1211inex1 5323 . . . . . . . . . 10 (𝑟𝐴) ∈ V
1312a1i 11 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑟𝑅) → (𝑟𝐴) ∈ V)
14 elrest 17474 . . . . . . . . . 10 ((𝑅𝑉𝐴𝑋) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
1514ad2ant2r 747 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
16 xpeq1 5703 . . . . . . . . . . . 12 (𝑢 = (𝑟𝐴) → (𝑢 × 𝑣) = ((𝑟𝐴) × 𝑣))
1716eqeq2d 2746 . . . . . . . . . . 11 (𝑢 = (𝑟𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × 𝑣)))
1817rexbidv 3177 . . . . . . . . . 10 (𝑢 = (𝑟𝐴) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣)))
19 vex 3482 . . . . . . . . . . . . 13 𝑠 ∈ V
2019inex1 5323 . . . . . . . . . . . 12 (𝑠𝐵) ∈ V
2120a1i 11 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑠𝑆) → (𝑠𝐵) ∈ V)
22 elrest 17474 . . . . . . . . . . . 12 ((𝑆𝑊𝐵𝑌) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
2322ad2ant2l 746 . . . . . . . . . . 11 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
24 xpeq2 5710 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐵) → ((𝑟𝐴) × 𝑣) = ((𝑟𝐴) × (𝑠𝐵)))
2524eqeq2d 2746 . . . . . . . . . . . 12 (𝑣 = (𝑠𝐵) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2625adantl 481 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑣 = (𝑠𝐵)) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2721, 23, 26rexxfr2d 5417 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2818, 27sylan9bbr 510 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑢 = (𝑟𝐴)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2913, 15, 28rexxfr2d 5417 . . . . . . . 8 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3011, 19xpex 7772 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
3130rgen2w 3064 . . . . . . . . 9 𝑟𝑅𝑠𝑆 (𝑟 × 𝑠) ∈ V
32 eqid 2735 . . . . . . . . . 10 (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
33 ineq1 4221 . . . . . . . . . . . 12 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34 inxp 5845 . . . . . . . . . . . 12 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵))
3533, 34eqtrdi 2791 . . . . . . . . . . 11 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵)))
3635eqeq2d 2746 . . . . . . . . . 10 (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3732, 36rexrnmpo 7573 . . . . . . . . 9 (∀𝑟𝑅𝑠𝑆 (𝑟 × 𝑠) ∈ V → (∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3831, 37ax-mp 5 . . . . . . . 8 (∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵)))
3929, 38bitr4di 289 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
4010, 39bitr4d 282 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)))
4140eqabdv 2873 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)})
42 eqid 2735 . . . . . 6 (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
4342rnmpo 7566 . . . . 5 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)}
4441, 43eqtr4di 2793 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
4544fveq2d 6911 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (topGen‘(ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
464, 8, 453eqtr2d 2781 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
47 ovex 7464 . . 3 (𝑅t 𝐴) ∈ V
48 ovex 7464 . . 3 (𝑆t 𝐵) ∈ V
49 eqid 2735 . . . 4 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
5049txval 23588 . . 3 (((𝑅t 𝐴) ∈ V ∧ (𝑆t 𝐵) ∈ V) → ((𝑅t 𝐴) ×t (𝑆t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
5147, 48, 50mp2an 692 . 2 ((𝑅t 𝐴) ×t (𝑆t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
5246, 51eqtr4di 2793 1 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅t 𝐴) ×t (𝑆t 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cin 3962   × cxp 5687  ran crn 5690  cfv 6563  (class class class)co 7431  cmpo 7433  t crest 17467  topGenctg 17484   ×t ctx 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-rest 17469  df-topgen 17490  df-tx 23586
This theorem is referenced by:  txlly  23660  txnlly  23661  txkgen  23676  cnmpt2res  23701  xkoinjcn  23711  cnmpopc  24969  cnheiborlem  25000  lhop1lem  26067  cxpcn3  26806  raddcn  33890  cvmlift2lem6  35293  cvmlift2lem9  35296  cvmlift2lem12  35299
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