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Theorem txrest 22382
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 2738 . . . . . 6 ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
21txval 22315 . . . . 5 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
32adantr 484 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
43oveq1d 7185 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
51txbasex 22317 . . . 4 ((𝑅𝑉𝑆𝑊) → ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6 xpexg 7491 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ∈ V)
7 tgrest 21910 . . . 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 599 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (topGen‘(ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
9 elrest 16804 . . . . . . . 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 599 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
11 vex 3402 . . . . . . . . . . 11 𝑟 ∈ V
1211inex1 5185 . . . . . . . . . 10 (𝑟𝐴) ∈ V
1312a1i 11 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑟𝑅) → (𝑟𝐴) ∈ V)
14 elrest 16804 . . . . . . . . . 10 ((𝑅𝑉𝐴𝑋) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
1514ad2ant2r 747 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
16 xpeq1 5539 . . . . . . . . . . . 12 (𝑢 = (𝑟𝐴) → (𝑢 × 𝑣) = ((𝑟𝐴) × 𝑣))
1716eqeq2d 2749 . . . . . . . . . . 11 (𝑢 = (𝑟𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × 𝑣)))
1817rexbidv 3207 . . . . . . . . . 10 (𝑢 = (𝑟𝐴) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣)))
19 vex 3402 . . . . . . . . . . . . 13 𝑠 ∈ V
2019inex1 5185 . . . . . . . . . . . 12 (𝑠𝐵) ∈ V
2120a1i 11 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑠𝑆) → (𝑠𝐵) ∈ V)
22 elrest 16804 . . . . . . . . . . . 12 ((𝑆𝑊𝐵𝑌) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
2322ad2ant2l 746 . . . . . . . . . . 11 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
24 xpeq2 5546 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐵) → ((𝑟𝐴) × 𝑣) = ((𝑟𝐴) × (𝑠𝐵)))
2524eqeq2d 2749 . . . . . . . . . . . 12 (𝑣 = (𝑠𝐵) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2625adantl 485 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑣 = (𝑠𝐵)) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2721, 23, 26rexxfr2d 5278 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2818, 27sylan9bbr 514 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑢 = (𝑟𝐴)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2913, 15, 28rexxfr2d 5278 . . . . . . . 8 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3011, 19xpex 7494 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
3130rgen2w 3066 . . . . . . . . 9 𝑟𝑅𝑠𝑆 (𝑟 × 𝑠) ∈ V
32 eqid 2738 . . . . . . . . . 10 (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
33 ineq1 4096 . . . . . . . . . . . 12 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34 inxp 5675 . . . . . . . . . . . 12 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵))
3533, 34eqtrdi 2789 . . . . . . . . . . 11 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵)))
3635eqeq2d 2749 . . . . . . . . . 10 (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3732, 36rexrnmpo 7305 . . . . . . . . 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 292 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
4010, 39bitr4d 285 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)))
4140abbi2dv 2869 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)})
42 eqid 2738 . . . . . 6 (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
4342rnmpo 7299 . . . . 5 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)}
4441, 43eqtr4di 2791 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
4544fveq2d 6678 . . 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 2779 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
47 ovex 7203 . . 3 (𝑅t 𝐴) ∈ V
48 ovex 7203 . . 3 (𝑆t 𝐵) ∈ V
49 eqid 2738 . . . 4 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
5049txval 22315 . . 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 2791 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 209  wa 399   = wceq 1542  wcel 2114  {cab 2716  wral 3053  wrex 3054  Vcvv 3398  cin 3842   × cxp 5523  ran crn 5526  cfv 6339  (class class class)co 7170  cmpo 7172  t crest 16797  topGenctg 16814   ×t ctx 22311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-rest 16799  df-topgen 16820  df-tx 22313
This theorem is referenced by:  txlly  22387  txnlly  22388  txkgen  22403  cnmpt2res  22428  xkoinjcn  22438  cnmpopc  23680  cnheiborlem  23706  lhop1lem  24765  cxpcn3  25489  raddcn  31451  cvmlift2lem6  32841  cvmlift2lem9  32844  cvmlift2lem12  32847
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