Theorem qliftlem 8092

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 8032	. . 3 ⊢ (𝑅 Er 𝑋 → (𝑋 ∈ V → 𝑅 ∈ V))
4	1, 2, 3	sylc 65	. 2 ⊢ (𝜑 → 𝑅 ∈ V)
5		ecelqsg 8066	. 2 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6	4, 5	sylan 577	1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3413  ⟨cop 4402   ↦ cmpt 4951  ran crn 5342   Er wer 8005  [cec 8006   / cqs 8007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-xp 5347  df-rel 5348  df-cnv 5349  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-er 8008  df-ec 8010  df-qs 8014

This theorem is referenced by:  qliftrel  8093  qliftel  8094  qliftel1  8095  qliftfun  8096  qliftf  8099  qliftval  8100