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Theorem txrest 21344
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 2621 . . . . . 6 ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
21txval 21277 . . . . 5 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
32adantr 481 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
43oveq1d 6619 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
51txbasex 21279 . . . 4 ((𝑅𝑉𝑆𝑊) → ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6 xpexg 6913 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ∈ V)
7 tgrest 20873 . . . 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 494 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (topGen‘(ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
9 elrest 16009 . . . . . . . 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 494 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
11 vex 3189 . . . . . . . . . . 11 𝑟 ∈ V
1211inex1 4759 . . . . . . . . . 10 (𝑟𝐴) ∈ V
1312a1i 11 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑟𝑅) → (𝑟𝐴) ∈ V)
14 elrest 16009 . . . . . . . . . 10 ((𝑅𝑉𝐴𝑋) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
1514ad2ant2r 782 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
16 xpeq1 5088 . . . . . . . . . . . 12 (𝑢 = (𝑟𝐴) → (𝑢 × 𝑣) = ((𝑟𝐴) × 𝑣))
1716eqeq2d 2631 . . . . . . . . . . 11 (𝑢 = (𝑟𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × 𝑣)))
1817rexbidv 3045 . . . . . . . . . 10 (𝑢 = (𝑟𝐴) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣)))
19 vex 3189 . . . . . . . . . . . . 13 𝑠 ∈ V
2019inex1 4759 . . . . . . . . . . . 12 (𝑠𝐵) ∈ V
2120a1i 11 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑠𝑆) → (𝑠𝐵) ∈ V)
22 elrest 16009 . . . . . . . . . . . 12 ((𝑆𝑊𝐵𝑌) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
2322ad2ant2l 781 . . . . . . . . . . 11 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
24 xpeq2 5089 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐵) → ((𝑟𝐴) × 𝑣) = ((𝑟𝐴) × (𝑠𝐵)))
2524eqeq2d 2631 . . . . . . . . . . . 12 (𝑣 = (𝑠𝐵) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2625adantl 482 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑣 = (𝑠𝐵)) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2721, 23, 26rexxfr2d 4843 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2818, 27sylan9bbr 736 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑢 = (𝑟𝐴)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2913, 15, 28rexxfr2d 4843 . . . . . . . 8 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3011, 19xpex 6915 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
3130rgen2w 2920 . . . . . . . . 9 𝑟𝑅𝑠𝑆 (𝑟 × 𝑠) ∈ V
32 eqid 2621 . . . . . . . . . 10 (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
33 ineq1 3785 . . . . . . . . . . . 12 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34 inxp 5214 . . . . . . . . . . . 12 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵))
3533, 34syl6eq 2671 . . . . . . . . . . 11 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵)))
3635eqeq2d 2631 . . . . . . . . . 10 (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3732, 36rexrnmpt2 6729 . . . . . . . . 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, 38syl6bbr 278 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
4010, 39bitr4d 271 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)))
4140abbi2dv 2739 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)})
42 eqid 2621 . . . . . 6 (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
4342rnmpt2 6723 . . . . 5 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)}
4441, 43syl6eqr 2673 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
4544fveq2d 6152 . . 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 2661 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
47 ovex 6632 . . 3 (𝑅t 𝐴) ∈ V
48 ovex 6632 . . 3 (𝑆t 𝐵) ∈ V
49 eqid 2621 . . . 4 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
5049txval 21277 . . 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 707 . 2 ((𝑅t 𝐴) ×t (𝑆t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
5246, 51syl6eqr 2673 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 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  wrex 2908  Vcvv 3186  cin 3554   × cxp 5072  ran crn 5075  cfv 5847  (class class class)co 6604  cmpt2 6606  t crest 16002  topGenctg 16019   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-rest 16004  df-topgen 16025  df-tx 21275
This theorem is referenced by:  txlly  21349  txnlly  21350  txkgen  21365  cnmpt2res  21390  xkoinjcn  21400  cnmpt2pc  22635  cnheiborlem  22661  lhop1lem  23680  cxpcn3  24389  raddcn  29757  cvmlift2lem6  30998  cvmlift2lem9  31001  cvmlift2lem12  31004
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