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Theorem qliftel 6274
Description: Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2 ((𝜑𝑥𝑋) → 𝐴𝑌)
qlift.3 (𝜑𝑅 Er 𝑋)
qlift.4 (𝜑𝑋 ∈ V)
Assertion
Ref Expression
qliftel (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴)))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem qliftel
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
3 qlift.3 . . . 4 (𝜑𝑅 Er 𝑋)
4 qlift.4 . . . 4 (𝜑𝑋 ∈ V)
51, 2, 3, 4qliftlem 6272 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftel 5485 . 2 (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
73adantr 270 . . . . 5 ((𝜑𝑥𝑋) → 𝑅 Er 𝑋)
8 simpr 108 . . . . 5 ((𝜑𝑥𝑋) → 𝑥𝑋)
97, 8erth2 6239 . . . 4 ((𝜑𝑥𝑋) → (𝐶𝑅𝑥 ↔ [𝐶]𝑅 = [𝑥]𝑅))
109anbi1d 453 . . 3 ((𝜑𝑥𝑋) → ((𝐶𝑅𝑥𝐷 = 𝐴) ↔ ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
1110rexbidva 2370 . 2 (𝜑 → (∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴) ↔ ∃𝑥𝑋 ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
126, 11bitr4d 189 1 (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 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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