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Theorem txbasex 22102
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 2818 . . . 4 𝑅 = 𝑅
3 eqid 2818 . . . 4 𝑆 = 𝑆
41, 2, 3txuni2 22101 . . 3 ( 𝑅 × 𝑆) = 𝐵
5 uniexg 7456 . . . 4 (𝑅𝑉 𝑅 ∈ V)
6 uniexg 7456 . . . 4 (𝑆𝑊 𝑆 ∈ V)
7 xpexg 7462 . . . 4 (( 𝑅 ∈ V ∧ 𝑆 ∈ V) → ( 𝑅 × 𝑆) ∈ V)
85, 6, 7syl2an 595 . . 3 ((𝑅𝑉𝑆𝑊) → ( 𝑅 × 𝑆) ∈ V)
94, 8eqeltrrid 2915 . 2 ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
10 uniexb 7475 . 2 (𝐵 ∈ V ↔ 𝐵 ∈ V)
119, 10sylibr 235 1 ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492   cuni 4830   × cxp 5546  ran crn 5549  cmpo 7147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679
This theorem is referenced by:  txbas  22103  eltx  22104  txtopon  22127  txopn  22138  txss12  22141  txbasval  22142  txrest  22167  sxsiga  31349  elsx  31352  mbfmco2  31422
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