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Theorem txss12 23419
Description: Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
txss12 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))

Proof of Theorem txss12
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2724 . . . 4 ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
21txbasex 23380 . . 3 ((𝐵𝑉𝐷𝑊) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
3 resmpo 7520 . . . . . 6 ((𝐴𝐵𝐶𝐷) → ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)))
4 resss 5996 . . . . . 6 ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
53, 4eqsstrrdi 4029 . . . . 5 ((𝐴𝐵𝐶𝐷) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
65adantl 481 . . . 4 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
7 rnss 5928 . . . 4 ((𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) → ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
86, 7syl 17 . . 3 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
9 tgss 22781 . . 3 ((ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V ∧ ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))) → (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
102, 8, 9syl2an2r 682 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
11 ssexg 5313 . . . . 5 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
12 ssexg 5313 . . . . 5 ((𝐶𝐷𝐷𝑊) → 𝐶 ∈ V)
13 eqid 2724 . . . . . 6 ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))
1413txval 23378 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1511, 12, 14syl2an 595 . . . 4 (((𝐴𝐵𝐵𝑉) ∧ (𝐶𝐷𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1615an4s 657 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝑉𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1716ancoms 458 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
181txval 23378 . . 3 ((𝐵𝑉𝐷𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
1918adantr 480 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2010, 17, 193sstr4d 4021 1 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  wss 3940   × cxp 5664  ran crn 5667  cres 5668  cfv 6533  (class class class)co 7401  cmpo 7403  topGenctg 17379   ×t ctx 23374
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 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
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 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-topgen 17385  df-tx 23376
This theorem is referenced by: (None)
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