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Mirrors > Home > MPE Home > Th. List > qliftlem | Structured version Visualization version GIF version |
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 8701 | . . 3 ⊢ (𝑅 Er 𝑋 → (𝑋 ∈ 𝑉 → 𝑅 ∈ V)) | |
4 | 1, 2, 3 | sylc 65 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) |
5 | ecelqsg 8740 | . 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 396 = wceq 1541 ∈ wcel 2106 Vcvv 3466 〈cop 4619 ↦ cmpt 5215 ran crn 5661 Er wer 8674 [cec 8675 / cqs 8676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5283 ax-nul 5290 ax-pow 5347 ax-pr 5411 ax-un 7699 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3426 df-v 3468 df-dif 3938 df-un 3940 df-in 3942 df-ss 3952 df-nul 4310 df-if 4514 df-pw 4589 df-sn 4614 df-pr 4616 df-op 4620 df-uni 4893 df-br 5133 df-opab 5195 df-xp 5666 df-rel 5667 df-cnv 5668 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-er 8677 df-ec 8679 df-qs 8683 |
This theorem is referenced by: qliftrel 8767 qliftel 8768 qliftel1 8769 qliftfun 8770 qliftf 8773 qliftval 8774 |
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