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Theorem txss12 21737
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 2799 . . . . 5 ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
21txbasex 21698 . . . 4 ((𝐵𝑉𝐷𝑊) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
32adantr 473 . . 3 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
4 resmpt2 6992 . . . . . 6 ((𝐴𝐵𝐶𝐷) → ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)))
5 resss 5632 . . . . . 6 ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
64, 5syl6eqssr 3852 . . . . 5 ((𝐴𝐵𝐶𝐷) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
76adantl 474 . . . 4 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
8 rnss 5557 . . . 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 21101 . . 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 580 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
12 ssexg 4999 . . . . 5 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
13 ssexg 4999 . . . . 5 ((𝐶𝐷𝐷𝑊) → 𝐶 ∈ V)
14 eqid 2799 . . . . . 6 ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))
1514txval 21696 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1612, 13, 15syl2an 590 . . . 4 (((𝐴𝐵𝐵𝑉) ∧ (𝐶𝐷𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1716an4s 651 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝑉𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1817ancoms 451 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
191txval 21696 . . 3 ((𝐵𝑉𝐷𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2019adantr 473 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2111, 18, 203sstr4d 3844 1 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  wss 3769   × cxp 5310  ran crn 5313  cres 5314  cfv 6101  (class class class)co 6878  cmpt2 6880  topGenctg 16413   ×t ctx 21692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-topgen 16419  df-tx 21694
This theorem is referenced by: (None)
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