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Mirrors > Home > ILE Home > Th. List > qliftlem | 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 6461 | . . 3 ⊢ (𝑅 Er 𝑋 → (𝑋 ∈ V → 𝑅 ∈ V)) | |
4 | 1, 2, 3 | sylc 62 | . 2 ⊢ (𝜑 → 𝑅 ∈ V) |
5 | ecelqsg 6490 | . 2 ⊢ ((𝑅 ∈ V ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) | |
6 | 4, 5 | sylan 281 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 Vcvv 2689 〈cop 3535 ↦ cmpt 3997 ran crn 4548 Er wer 6434 [cec 6435 / cqs 6436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-rel 4554 df-cnv 4555 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-er 6437 df-ec 6439 df-qs 6443 |
This theorem is referenced by: qliftrel 6516 qliftel 6517 qliftel1 6518 qliftfun 6519 qliftf 6522 qliftval 6523 |
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