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Theorem txbasex 21274
Description: The basis for the product topology is a set. (Contributed by Mario Carneiro, 2-Sep-2015.)
Hypothesis
Ref Expression
txval.1 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
Assertion
Ref Expression
txbasex ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem txbasex
StepHypRef Expression
1 txval.1 . . . 4 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
2 eqid 2626 . . . 4 𝑅 = 𝑅
3 eqid 2626 . . . 4 𝑆 = 𝑆
41, 2, 3txuni2 21273 . . 3 ( 𝑅 × 𝑆) = 𝐵
5 uniexg 6909 . . . 4 (𝑅𝑉 𝑅 ∈ V)
6 uniexg 6909 . . . 4 (𝑆𝑊 𝑆 ∈ V)
7 xpexg 6914 . . . 4 (( 𝑅 ∈ V ∧ 𝑆 ∈ V) → ( 𝑅 × 𝑆) ∈ V)
85, 6, 7syl2an 494 . . 3 ((𝑅𝑉𝑆𝑊) → ( 𝑅 × 𝑆) ∈ V)
94, 8syl5eqelr 2709 . 2 ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
10 uniexb 6922 . 2 (𝐵 ∈ V ↔ 𝐵 ∈ V)
119, 10sylibr 224 1 ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  Vcvv 3191   cuni 4407   × cxp 5077  ran crn 5080  cmpt2 6607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-oprab 6609  df-mpt2 6610  df-1st 7116  df-2nd 7117
This theorem is referenced by:  txbas  21275  eltx  21276  txtopon  21299  txopn  21310  txss12  21313  txbasval  21314  txrest  21339  sxsiga  30027  elsx  30030  mbfmco2  30100
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