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Theorem txbasex 22180
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 2824 . . . 4 𝑅 = 𝑅
3 eqid 2824 . . . 4 𝑆 = 𝑆
41, 2, 3txuni2 22179 . . 3 ( 𝑅 × 𝑆) = 𝐵
5 uniexg 7462 . . . 4 (𝑅𝑉 𝑅 ∈ V)
6 uniexg 7462 . . . 4 (𝑆𝑊 𝑆 ∈ V)
7 xpexg 7469 . . . 4 (( 𝑅 ∈ V ∧ 𝑆 ∈ V) → ( 𝑅 × 𝑆) ∈ V)
85, 6, 7syl2an 598 . . 3 ((𝑅𝑉𝑆𝑊) → ( 𝑅 × 𝑆) ∈ V)
94, 8eqeltrrid 2921 . 2 ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
10 uniexb 7482 . 2 (𝐵 ∈ V ↔ 𝐵 ∈ V)
119, 10sylibr 237 1 ((𝑅𝑉𝑆𝑊) → 𝐵 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480   cuni 4824   × cxp 5541  ran crn 5544  cmpo 7153
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-fv 6353  df-oprab 7155  df-mpo 7156  df-1st 7686  df-2nd 7687
This theorem is referenced by:  txbas  22181  eltx  22182  txtopon  22205  txopn  22216  txss12  22219  txbasval  22220  txrest  22245  sxsiga  31535  elsx  31538  mbfmco2  31608
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