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Theorem txrest 12287
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 2115 . . . . . 6 ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
21txval 12266 . . . . 5 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
32adantr 272 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))))
43oveq1d 5743 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
51txbasex 12268 . . . 4 ((𝑅𝑉𝑆𝑊) → ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ∈ V)
6 xpexg 4613 . . . 4 ((𝐴𝑋𝐵𝑌) → (𝐴 × 𝐵) ∈ V)
7 tgrest 12181 . . . 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 285 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (topGen‘(ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵))) = ((topGen‘ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))) ↾t (𝐴 × 𝐵)))
9 elrest 11970 . . . . . . . 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 285 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
11 vex 2660 . . . . . . . . . . 11 𝑟 ∈ V
1211inex1 4022 . . . . . . . . . 10 (𝑟𝐴) ∈ V
1312a1i 9 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑟𝑅) → (𝑟𝐴) ∈ V)
14 elrest 11970 . . . . . . . . . 10 ((𝑅𝑉𝐴𝑋) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
1514ad2ant2r 498 . . . . . . . . 9 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑢 ∈ (𝑅t 𝐴) ↔ ∃𝑟𝑅 𝑢 = (𝑟𝐴)))
16 xpeq1 4513 . . . . . . . . . . . 12 (𝑢 = (𝑟𝐴) → (𝑢 × 𝑣) = ((𝑟𝐴) × 𝑣))
1716eqeq2d 2126 . . . . . . . . . . 11 (𝑢 = (𝑟𝐴) → (𝑥 = (𝑢 × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × 𝑣)))
1817rexbidv 2412 . . . . . . . . . 10 (𝑢 = (𝑟𝐴) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣)))
19 vex 2660 . . . . . . . . . . . . 13 𝑠 ∈ V
2019inex1 4022 . . . . . . . . . . . 12 (𝑠𝐵) ∈ V
2120a1i 9 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑠𝑆) → (𝑠𝐵) ∈ V)
22 elrest 11970 . . . . . . . . . . . 12 ((𝑆𝑊𝐵𝑌) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
2322ad2ant2l 497 . . . . . . . . . . 11 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑣 ∈ (𝑆t 𝐵) ↔ ∃𝑠𝑆 𝑣 = (𝑠𝐵)))
24 xpeq2 4514 . . . . . . . . . . . . 13 (𝑣 = (𝑠𝐵) → ((𝑟𝐴) × 𝑣) = ((𝑟𝐴) × (𝑠𝐵)))
2524eqeq2d 2126 . . . . . . . . . . . 12 (𝑣 = (𝑠𝐵) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2625adantl 273 . . . . . . . . . . 11 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑣 = (𝑠𝐵)) → (𝑥 = ((𝑟𝐴) × 𝑣) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2721, 23, 26rexxfr2d 4346 . . . . . . . . . 10 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = ((𝑟𝐴) × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2818, 27sylan9bbr 456 . . . . . . . . 9 ((((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) ∧ 𝑢 = (𝑟𝐴)) → (∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
2913, 15, 28rexxfr2d 4346 . . . . . . . 8 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3011, 19xpex 4614 . . . . . . . . . 10 (𝑟 × 𝑠) ∈ V
3130rgen2w 2462 . . . . . . . . 9 𝑟𝑅𝑠𝑆 (𝑟 × 𝑠) ∈ V
32 eqid 2115 . . . . . . . . . 10 (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) = (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))
33 ineq1 3236 . . . . . . . . . . . 12 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)))
34 inxp 4633 . . . . . . . . . . . 12 ((𝑟 × 𝑠) ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵))
3533, 34syl6eq 2163 . . . . . . . . . . 11 (𝑤 = (𝑟 × 𝑠) → (𝑤 ∩ (𝐴 × 𝐵)) = ((𝑟𝐴) × (𝑠𝐵)))
3635eqeq2d 2126 . . . . . . . . . 10 (𝑤 = (𝑟 × 𝑠) → (𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3732, 36rexrnmpo 5840 . . . . . . . . 9 (∀𝑟𝑅𝑠𝑆 (𝑟 × 𝑠) ∈ V → (∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵))))
3831, 37ax-mp 7 . . . . . . . 8 (∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵)) ↔ ∃𝑟𝑅𝑠𝑆 𝑥 = ((𝑟𝐴) × (𝑠𝐵)))
3929, 38syl6bbr 197 . . . . . . 7 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣) ↔ ∃𝑤 ∈ ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠))𝑥 = (𝑤 ∩ (𝐴 × 𝐵))))
4010, 39bitr4d 190 . . . . . 6 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑥 ∈ (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) ↔ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)))
4140abbi2dv 2233 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)})
42 eqid 2115 . . . . . 6 (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
4342rnmpo 5835 . . . . 5 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = {𝑥 ∣ ∃𝑢 ∈ (𝑅t 𝐴)∃𝑣 ∈ (𝑆t 𝐵)𝑥 = (𝑢 × 𝑣)}
4441, 43syl6eqr 2165 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (ran (𝑟𝑅, 𝑠𝑆 ↦ (𝑟 × 𝑠)) ↾t (𝐴 × 𝐵)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)))
4544fveq2d 5379 . . 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 2153 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
47 restfn 11967 . . . 4 t Fn (V × V)
48 simpll 501 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝑅𝑉)
4948elexd 2670 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝑅 ∈ V)
50 simprl 503 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴𝑋)
5150elexd 2670 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝐴 ∈ V)
52 fnovex 5758 . . . 4 (( ↾t Fn (V × V) ∧ 𝑅 ∈ V ∧ 𝐴 ∈ V) → (𝑅t 𝐴) ∈ V)
5347, 49, 51, 52mp3an2i 1303 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑅t 𝐴) ∈ V)
54 simplr 502 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝑆𝑊)
5554elexd 2670 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝑆 ∈ V)
56 simprr 504 . . . . 5 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵𝑌)
5756elexd 2670 . . . 4 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → 𝐵 ∈ V)
58 fnovex 5758 . . . 4 (( ↾t Fn (V × V) ∧ 𝑆 ∈ V ∧ 𝐵 ∈ V) → (𝑆t 𝐵) ∈ V)
5947, 55, 57, 58mp3an2i 1303 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → (𝑆t 𝐵) ∈ V)
60 eqid 2115 . . . 4 ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))
6160txval 12266 . . 3 (((𝑅t 𝐴) ∈ V ∧ (𝑆t 𝐵) ∈ V) → ((𝑅t 𝐴) ×t (𝑆t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
6253, 59, 61syl2anc 406 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅t 𝐴) ×t (𝑆t 𝐵)) = (topGen‘ran (𝑢 ∈ (𝑅t 𝐴), 𝑣 ∈ (𝑆t 𝐵) ↦ (𝑢 × 𝑣))))
6346, 62eqtr4d 2150 1 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑋𝐵𝑌)) → ((𝑅 ×t 𝑆) ↾t (𝐴 × 𝐵)) = ((𝑅t 𝐴) ×t (𝑆t 𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  {cab 2101  wral 2390  wrex 2391  Vcvv 2657  cin 3036   × cxp 4497  ran crn 4500   Fn wfn 5076  cfv 5081  (class class class)co 5728  cmpo 5730  t crest 11963  topGenctg 11978   ×t ctx 12263
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-rest 11965  df-topgen 11984  df-tx 12264
This theorem is referenced by:  cnmpt2res  12308  limccnp2cntop  12602
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