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Mirrors > Home > MPE Home > Th. List > qliftlem | Structured version Visualization version GIF version |
Description: 𝐹, a function lift, is a subset of 𝑅 × 𝑆. (Contributed by Mario Carneiro, 23-Dec-2016.) |
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 8312 | . . 3 ⊢ (𝑅 Er 𝑋 → (𝑋 ∈ V → 𝑅 ∈ V)) | |
4 | 1, 2, 3 | sylc 65 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) |
5 | ecelqsg 8351 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) | |
6 | 4, 5 | sylan 582 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4572 ↦ cmpt 5145 ran crn 5555 Er wer 8285 [cec 8286 / cqs 8287 |
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 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-er 8288 df-ec 8290 df-qs 8294 |
This theorem is referenced by: qliftrel 8378 qliftel 8379 qliftel1 8380 qliftfun 8381 qliftf 8384 qliftval 8385 |
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