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| Description: Lemma for theorems about a function lift. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.) | 
| 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) | 
| Ref | Expression | 
|---|---|
| qliftlem | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | qlift.3 | . . 3 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
| 2 | qlift.4 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
| 3 | erex 8769 | . . 3 ⊢ (𝑅 Er 𝑋 → (𝑋 ∈ 𝑉 → 𝑅 ∈ V)) | |
| 4 | 1, 2, 3 | sylc 65 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) | 
| 5 | ecelqsg 8812 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) | |
| 6 | 4, 5 | sylan 580 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↦ cmpt 5225 ran crn 5686 Er wer 8742 [cec 8743 / cqs 8744 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-er 8745 df-ec 8747 df-qs 8751 | 
| This theorem is referenced by: qliftrel 8839 qliftel 8840 qliftel1 8841 qliftfun 8842 qliftf 8845 qliftval 8846 | 
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