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Theorem txss12 21456
 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 2651 . . . . 5 ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
21txbasex 21417 . . . 4 ((𝐵𝑉𝐷𝑊) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
32adantr 480 . . 3 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
4 resmpt2 6800 . . . . . 6 ((𝐴𝐵𝐶𝐷) → ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)))
5 resss 5457 . . . . . 6 ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
64, 5syl6eqssr 3689 . . . . 5 ((𝐴𝐵𝐶𝐷) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
76adantl 481 . . . 4 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
8 rnss 5386 . . . 4 ((𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) → ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
97, 8syl 17 . . 3 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
10 tgss 20820 . . 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))))
113, 9, 10syl2anc 694 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
12 ssexg 4837 . . . . 5 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
13 ssexg 4837 . . . . 5 ((𝐶𝐷𝐷𝑊) → 𝐶 ∈ V)
14 eqid 2651 . . . . . 6 ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))
1514txval 21415 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1612, 13, 15syl2an 493 . . . 4 (((𝐴𝐵𝐵𝑉) ∧ (𝐶𝐷𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1716an4s 886 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝑉𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1817ancoms 468 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
191txval 21415 . . 3 ((𝐵𝑉𝐷𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2019adantr 480 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2111, 18, 203sstr4d 3681 1 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  Vcvv 3231   ⊆ wss 3607   × cxp 5141  ran crn 5144   ↾ cres 5145  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  topGenctg 16145   ×t ctx 21411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-topgen 16151  df-tx 21413 This theorem is referenced by: (None)
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