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Theorem txrest 21714
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 2765 . . . . . 6 ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
21txval 21647 . . . . 5 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
32adantr 472 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
43oveq1d 6857 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
51txbasex 21649 . . . 4 ((𝑅𝑉𝑆𝑊) → ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6 xpexg 7158 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ∈ V)
7 tgrest 21243 . . . 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 589 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (topGen‘(ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
9 elrest 16354 . . . . . . . 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 589 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
11 vex 3353 . . . . . . . . . . 11 𝑟 ∈ V
1211inex1 4960 . . . . . . . . . 10 (𝑟𝐴) ∈ V
1312a1i 11 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑟𝑅) → (𝑟𝐴) ∈ V)
14 elrest 16354 . . . . . . . . . 10 ((𝑅𝑉𝐴𝑋) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
1514ad2ant2r 753 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
16 xpeq1 5291 . . . . . . . . . . . 12 (𝑢 = (𝑟𝐴) → (𝑢 × 𝑣) = ((𝑟𝐴) × 𝑣))
1716eqeq2d 2775 . . . . . . . . . . 11 (𝑢 = (𝑟𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × 𝑣)))
1817rexbidv 3199 . . . . . . . . . 10 (𝑢 = (𝑟𝐴) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣)))
19 vex 3353 . . . . . . . . . . . . 13 𝑠 ∈ V
2019inex1 4960 . . . . . . . . . . . 12 (𝑠𝐵) ∈ V
2120a1i 11 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑠𝑆) → (𝑠𝐵) ∈ V)
22 elrest 16354 . . . . . . . . . . . 12 ((𝑆𝑊𝐵𝑌) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
2322ad2ant2l 752 . . . . . . . . . . 11 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
24 xpeq2 5298 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐵) → ((𝑟𝐴) × 𝑣) = ((𝑟𝐴) × (𝑠𝐵)))
2524eqeq2d 2775 . . . . . . . . . . . 12 (𝑣 = (𝑠𝐵) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2625adantl 473 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑣 = (𝑠𝐵)) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2721, 23, 26rexxfr2d 5046 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2818, 27sylan9bbr 506 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑢 = (𝑟𝐴)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2913, 15, 28rexxfr2d 5046 . . . . . . . 8 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3011, 19xpex 7160 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
3130rgen2w 3072 . . . . . . . . 9 𝑟𝑅𝑠𝑆 (𝑟 × 𝑠) ∈ V
32 eqid 2765 . . . . . . . . . 10 (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
33 ineq1 3969 . . . . . . . . . . . 12 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34 inxp 5423 . . . . . . . . . . . 12 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵))
3533, 34syl6eq 2815 . . . . . . . . . . 11 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵)))
3635eqeq2d 2775 . . . . . . . . . 10 (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3732, 36rexrnmpt2 6974 . . . . . . . . 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 280 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
4010, 39bitr4d 273 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)))
4140abbi2dv 2885 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)})
42 eqid 2765 . . . . . 6 (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
4342rnmpt2 6968 . . . . 5 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)}
4441, 43syl6eqr 2817 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
4544fveq2d 6379 . . 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 2805 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
47 ovex 6874 . . 3 (𝑅t 𝐴) ∈ V
48 ovex 6874 . . 3 (𝑆t 𝐵) ∈ V
49 eqid 2765 . . . 4 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
5049txval 21647 . . 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 683 . 2 ((𝑅t 𝐴) ×t (𝑆t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
5246, 51syl6eqr 2817 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 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  wral 3055  wrex 3056  Vcvv 3350  cin 3731   × cxp 5275  ran crn 5278  cfv 6068  (class class class)co 6842  cmpt2 6844  t crest 16347  topGenctg 16364   ×t ctx 21643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-rest 16349  df-topgen 16370  df-tx 21645
This theorem is referenced by:  txlly  21719  txnlly  21720  txkgen  21735  cnmpt2res  21760  xkoinjcn  21770  cnmpt2pc  23006  cnheiborlem  23032  lhop1lem  24067  cxpcn3  24780  raddcn  30422  cvmlift2lem6  31738  cvmlift2lem9  31741  cvmlift2lem12  31744
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