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Theorem qliftel1 8397
 Description: Elementhood in the relation 𝐹. (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
qliftel1 ((𝜑𝑥𝑋) → [𝑥]𝑅𝐹𝐴)
Distinct variable groups:   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem qliftel1
StepHypRef Expression
1 qlift.1 . 2 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 qlift.2 . . 3 ((𝜑𝑥𝑋) → 𝐴𝑌)
3 qlift.3 . . 3 (𝜑𝑅 Er 𝑋)
4 qlift.4 . . 3 (𝜑𝑋𝑉)
51, 2, 3, 4qliftlem 8394 . 2 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftel1 7063 1 ((𝜑𝑥𝑋) → [𝑥]𝑅𝐹𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   class class class wbr 5036   ↦ cmpt 5116  ran crn 5529   Er wer 8302  [cec 8303   / cqs 8304 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-xp 5534  df-rel 5535  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-er 8305  df-ec 8307  df-qs 8311 This theorem is referenced by: (None)
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