Theorem qliftlem 8545

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 (𝑥 ∈ 𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
qlift.3	⊢ (𝜑 → 𝑅 Er 𝑋)
qlift.4	⊢ (𝜑 → 𝑋 ∈ 𝑉)

Assertion

Ref	Expression
qliftlem	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [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

Step	Hyp	Ref	Expression
1		qlift.3	. . 3 ⊢ (𝜑 → 𝑅 Er 𝑋)
2		qlift.4	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
3		erex 8480	. . 3 ⊢ (𝑅 Er 𝑋 → (𝑋 ∈ 𝑉 → 𝑅 ∈ V))
4	1, 2, 3	sylc 65	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
5		ecelqsg 8519	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	4, 5	sylan 579	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564   ↦ cmpt 5153  ran crn 5581   Er wer 8453  [cec 8454   / cqs 8455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-er 8456  df-ec 8458  df-qs 8462

This theorem is referenced by:  qliftrel  8546  qliftel  8547  qliftel1  8548  qliftfun  8549  qliftf  8552  qliftval  8553