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Theorem txss12 23100
Description: Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
txss12 (((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷)) β†’ (𝐴 Γ—t 𝐢) βŠ† (𝐡 Γ—t 𝐷))
Proof of Theorem txss12
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . 4 ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))
21txbasex 23061 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) β†’ ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) ∈ V)
3 resmpo 7524 . . . . . 6 ((𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷) β†’ ((π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) β†Ύ (𝐴 Γ— 𝐢)) = (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)))
4 resss 6004 . . . . . 6 ((π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) β†Ύ (𝐴 Γ— 𝐢)) βŠ† (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))
53, 4eqsstrrdi 4036 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷) β†’ (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) βŠ† (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)))
65adantl 482 . . . 4 (((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷)) β†’ (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) βŠ† (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)))
7 rnss 5936 . . . 4 ((π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) βŠ† (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) β†’ ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) βŠ† ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)))
86, 7syl 17 . . 3 (((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷)) β†’ ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) βŠ† ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)))
9 tgss 22462 . . 3 ((ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) ∈ V ∧ ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) βŠ† ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))) βŠ† (topGenβ€˜ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))))
102, 8, 9syl2an2r 683 . 2 (((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷)) β†’ (topGenβ€˜ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))) βŠ† (topGenβ€˜ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))))
11 ssexg 5322 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
12 ssexg 5322 . . . . 5 ((𝐢 βŠ† 𝐷 ∧ 𝐷 ∈ π‘Š) β†’ 𝐢 ∈ V)
13 eqid 2732 . . . . . 6 ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))
1413txval 23059 . . . . 5 ((𝐴 ∈ V ∧ 𝐢 ∈ V) β†’ (𝐴 Γ—t 𝐢) = (topGenβ€˜ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))))
1511, 12, 14syl2an 596 . . . 4 (((𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ 𝑉) ∧ (𝐢 βŠ† 𝐷 ∧ 𝐷 ∈ π‘Š)) β†’ (𝐴 Γ—t 𝐢) = (topGenβ€˜ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))))
1615an4s 658 . . 3 (((𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷) ∧ (𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š)) β†’ (𝐴 Γ—t 𝐢) = (topGenβ€˜ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))))
1716ancoms 459 . 2 (((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷)) β†’ (𝐴 Γ—t 𝐢) = (topGenβ€˜ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))))
181txval 23059 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) β†’ (𝐡 Γ—t 𝐷) = (topGenβ€˜ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))))
1918adantr 481 . 2 (((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷)) β†’ (𝐡 Γ—t 𝐷) = (topGenβ€˜ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))))
2010, 17, 193sstr4d 4028 1 (((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) ∧ (𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷)) β†’ (𝐴 Γ—t 𝐢) βŠ† (𝐡 Γ—t 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3947   Γ— cxp 5673  ran crn 5676   β†Ύ cres 5677  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  topGenctg 17379   Γ—t ctx 23055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-topgen 17385  df-tx 23057
This theorem is referenced by: (None)
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