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Theorem ecex2 35693
Description: Condition for a coset to be a set. (Contributed by Peter Mazsa, 4-May-2019.)
Assertion
Ref Expression
ecex2 ((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))

Proof of Theorem ecex2
StepHypRef Expression
1 ecexg 8289 . 2 ((𝑅𝐴) ∈ 𝑉 → [𝐵](𝑅𝐴) ∈ V)
2 ecres2 35644 . . 3 (𝐵𝐴 → [𝐵](𝑅𝐴) = [𝐵]𝑅)
32eleq1d 2900 . 2 (𝐵𝐴 → ([𝐵](𝑅𝐴) ∈ V ↔ [𝐵]𝑅 ∈ V))
41, 3syl5ibcom 248 1 ((𝑅𝐴) ∈ 𝑉 → (𝐵𝐴 → [𝐵]𝑅 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  Vcvv 3480  cres 5544  [cec 8283
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 5189  ax-nul 5196  ax-pr 5317  ax-un 7455
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-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ec 8287
This theorem is referenced by:  uniqsALTV  35694
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