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Mirrors > Home > ILE Home > Th. List > qliftel | GIF version |
Description: Elementhood in the relation 𝐹. (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 |
---|---|
qliftel | ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qlift.1 | . . 3 ⊢ 𝐹 = ran (𝑥 ∈ 𝑋 ↦ 〈[𝑥]𝑅, 𝐴〉) | |
2 | qlift.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌) | |
3 | qlift.3 | . . . 4 ⊢ (𝜑 → 𝑅 Er 𝑋) | |
4 | qlift.4 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ V) | |
5 | 1, 2, 3, 4 | qliftlem 6272 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅)) |
6 | 1, 5, 2 | fliftel 5485 | . 2 ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 ([𝐶]𝑅 = [𝑥]𝑅 ∧ 𝐷 = 𝐴))) |
7 | 3 | adantr 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑅 Er 𝑋) |
8 | simpr 108 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | |
9 | 7, 8 | erth2 6239 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐶𝑅𝑥 ↔ [𝐶]𝑅 = [𝑥]𝑅)) |
10 | 9 | anbi1d 453 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐶𝑅𝑥 ∧ 𝐷 = 𝐴) ↔ ([𝐶]𝑅 = [𝑥]𝑅 ∧ 𝐷 = 𝐴))) |
11 | 10 | rexbidva 2370 | . 2 ⊢ (𝜑 → (∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴) ↔ ∃𝑥 ∈ 𝑋 ([𝐶]𝑅 = [𝑥]𝑅 ∧ 𝐷 = 𝐴))) |
12 | 6, 11 | bitr4d 189 | 1 ⊢ (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥 ∈ 𝑋 (𝐶𝑅𝑥 ∧ 𝐷 = 𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 ∃wrex 2354 Vcvv 2610 〈cop 3419 class class class wbr 3805 ↦ cmpt 3859 ran crn 4392 Er wer 6191 [cec 6192 / cqs 6193 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2612 df-sbc 2825 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-br 3806 df-opab 3860 df-mpt 3861 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-er 6194 df-ec 6196 df-qs 6200 |
This theorem is referenced by: (None) |
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