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Theorem txss12 23460
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 2726 . . . 4 ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))
21txbasex 23421 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐷 ∈ π‘Š) β†’ ran (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) ∈ V)
3 resmpo 7523 . . . . . 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 5999 . . . . . 6 ((π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦)) β†Ύ (𝐴 Γ— 𝐢)) βŠ† (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐷 ↦ (π‘₯ Γ— 𝑦))
53, 4eqsstrrdi 4032 . . . . 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 5931 . . . 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 22822 . . 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 682 . 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 5316 . . . . 5 ((𝐴 βŠ† 𝐡 ∧ 𝐡 ∈ 𝑉) β†’ 𝐴 ∈ V)
12 ssexg 5316 . . . . 5 ((𝐢 βŠ† 𝐷 ∧ 𝐷 ∈ π‘Š) β†’ 𝐢 ∈ V)
13 eqid 2726 . . . . . 6 ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦)) = ran (π‘₯ ∈ 𝐴, 𝑦 ∈ 𝐢 ↦ (π‘₯ Γ— 𝑦))
1413txval 23419 . . . . 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 657 . . 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 23419 . . 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 4024 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 1533   ∈ wcel 2098  Vcvv 3468   βŠ† wss 3943   Γ— cxp 5667  ran crn 5670   β†Ύ cres 5671  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  topGenctg 17390   Γ—t ctx 23415
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-topgen 17396  df-tx 23417
This theorem is referenced by: (None)
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