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Theorem qliftlem 8587
Description: Lemma for theorems about a function lift. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2 ((𝜑𝑥𝑋) → 𝐴𝑌)
qlift.3 (𝜑𝑅 Er 𝑋)
qlift.4 (𝜑𝑋𝑉)
Assertion
Ref Expression
qliftlem ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
Distinct variable groups:   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem qliftlem
StepHypRef Expression
1 qlift.3 . . 3 (𝜑𝑅 Er 𝑋)
2 qlift.4 . . 3 (𝜑𝑋𝑉)
3 erex 8522 . . 3 (𝑅 Er 𝑋 → (𝑋𝑉𝑅 ∈ V))
41, 2, 3sylc 65 . 2 (𝜑𝑅 ∈ V)
5 ecelqsg 8561 . 2 ((𝑅 ∈ V ∧ 𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
64, 5sylan 580 1 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cop 4567  cmpt 5157  ran crn 5590   Er wer 8495  [cec 8496   / cqs 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-er 8498  df-ec 8500  df-qs 8504
This theorem is referenced by:  qliftrel  8588  qliftel  8589  qliftel1  8590  qliftfun  8591  qliftf  8594  qliftval  8595
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