Theorem qliftlem 8372

Description: 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.)

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)

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 8307	. . 3 ⊢ (𝑅 Er 𝑋 → (𝑋 ∈ V → 𝑅 ∈ V))
4	1, 2, 3	sylc 65	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
5		ecelqsg 8346	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	4, 5	sylan 582	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ⟨cop 4566   ↦ cmpt 5138  ran crn 5550   Er wer 8280  [cec 8281   / cqs 8282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-er 8283  df-ec 8285  df-qs 8289

This theorem is referenced by:  qliftrel  8373  qliftel  8374  qliftel1  8375  qliftfun  8376  qliftf  8379  qliftval  8380