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Theorem txss12 22305
 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 2758 . . . 4 ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
21txbasex 22266 . . 3 ((𝐵𝑉𝐷𝑊) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
3 resmpo 7266 . . . . . 6 ((𝐴𝐵𝐶𝐷) → ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)))
4 resss 5848 . . . . . 6 ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
53, 4eqsstrrdi 3947 . . . . 5 ((𝐴𝐵𝐶𝐷) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
65adantl 485 . . . 4 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
7 rnss 5780 . . . 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 21668 . . 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 684 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
11 ssexg 5193 . . . . 5 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
12 ssexg 5193 . . . . 5 ((𝐶𝐷𝐷𝑊) → 𝐶 ∈ V)
13 eqid 2758 . . . . . 6 ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))
1413txval 22264 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1511, 12, 14syl2an 598 . . . 4 (((𝐴𝐵𝐵𝑉) ∧ (𝐶𝐷𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1615an4s 659 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝑉𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1716ancoms 462 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
181txval 22264 . . 3 ((𝐵𝑉𝐷𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
1918adantr 484 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2010, 17, 193sstr4d 3939 1 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858   × cxp 5522  ran crn 5525   ↾ cres 5526  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152  topGenctg 16769   ×t ctx 22260 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-topgen 16775  df-tx 22262 This theorem is referenced by: (None)
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