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Theorem txss12 23522
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 2728 . . . 4 ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))
21txbasex 23483 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) β†’ ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) ∈ V)
3 resmpo 7540 . . . . . 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 6010 . . . . . 6 ((π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) β†Ύ (𝐴 Γ— 𝐢)) βŠ† (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))
53, 4eqsstrrdi 4035 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ 𝐢 βŠ† 𝐷) β†’ (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) βŠ† (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)))
65adantl 481 . . . 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 5941 . . . 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 22884 . . 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 684 . 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 5323 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
12 ssexg 5323 . . . . 5 ((𝐢 βŠ† 𝐷 ∧ 𝐷 ∈ π‘Š) β†’ 𝐢 ∈ V)
13 eqid 2728 . . . . . 6 ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))
1413txval 23481 . . . . 5 ((𝐴 ∈ V ∧ 𝐢 ∈ V) β†’ (𝐴 Γ—t 𝐢) = (topGenβ€˜ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))))
1511, 12, 14syl2an 595 . . . 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 659 . . 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 458 . 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 23481 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) β†’ (𝐡 Γ—t 𝐷) = (topGenβ€˜ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))))
1918adantr 480 . 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 4027 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 395   = wceq 1534   ∈ wcel 2099  Vcvv 3471   βŠ† wss 3947   Γ— cxp 5676  ran crn 5679   β†Ύ cres 5680  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422  topGenctg 17419   Γ—t ctx 23477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-topgen 17425  df-tx 23479
This theorem is referenced by: (None)
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