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Metamath Proof Explorer

Theorem qliftlem

Description: Lemma for theorems about a function lift. (Contributed by Mario Carneiro, 23-Dec-2016) (Revised by AV, 3-Aug-2024)

Ref Expression
Hypotheses 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 qliftlem ⊢ φ ∧ x ∈ X → x R ∈ X / R

Proof

Step Hyp Ref Expression
1 qlift.1 ⊢ F = ran ⁡ x ∈ X ⟼ x R A
2 qlift.2 ⊢ φ ∧ x ∈ X → A ∈ Y
3 qlift.3 ⊢ φ → R Er X
4 qlift.4 ⊢ φ → X ∈ V
5 erex ⊢ R Er X → X ∈ V → R ∈ V
6 3 4 5 sylc ⊢ φ → R ∈ V
7 ecelqsw ⊢ R ∈ V ∧ x ∈ X → x R ∈ X / R
8 6 7 sylan ⊢ φ ∧ x ∈ X → x R ∈ X / R