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Theorem txss12 21629
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 2771 . . . . 5 ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
21txbasex 21590 . . . 4 ((𝐵𝑉𝐷𝑊) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
32adantr 466 . . 3 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
4 resmpt2 6905 . . . . . 6 ((𝐴𝐵𝐶𝐷) → ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)))
5 resss 5563 . . . . . 6 ((𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))
64, 5syl6eqssr 3805 . . . . 5 ((𝐴𝐵𝐶𝐷) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
76adantl 467 . . . 4 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦)))
8 rnss 5492 . . . 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 20993 . . 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 573 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
12 ssexg 4938 . . . . 5 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
13 ssexg 4938 . . . . 5 ((𝐶𝐷𝐷𝑊) → 𝐶 ∈ V)
14 eqid 2771 . . . . . 6 ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))
1514txval 21588 . . . . 5 ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1612, 13, 15syl2an 583 . . . 4 (((𝐴𝐵𝐵𝑉) ∧ (𝐶𝐷𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1716an4s 639 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝑉𝐷𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
1817ancoms 455 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥𝐴, 𝑦𝐶 ↦ (𝑥 × 𝑦))))
191txval 21588 . . 3 ((𝐵𝑉𝐷𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2019adantr 466 . 2 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥𝐵, 𝑦𝐷 ↦ (𝑥 × 𝑦))))
2111, 18, 203sstr4d 3797 1 (((𝐵𝑉𝐷𝑊) ∧ (𝐴𝐵𝐶𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  wss 3723   × cxp 5247  ran crn 5250  cres 5251  cfv 6031  (class class class)co 6793  cmpt2 6795  topGenctg 16306   ×t ctx 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-topgen 16312  df-tx 21586
This theorem is referenced by: (None)
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