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Theorem qliftel 7790
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 7788 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftel 6524 . 2 (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
73adantr 481 . . . . 5 ((𝜑𝑥𝑋) → 𝑅 Er 𝑋)
8 simpr 477 . . . . 5 ((𝜑𝑥𝑋) → 𝑥𝑋)
97, 8erth2 7752 . . . 4 ((𝜑𝑥𝑋) → (𝐶𝑅𝑥 ↔ [𝐶]𝑅 = [𝑥]𝑅))
109anbi1d 740 . . 3 ((𝜑𝑥𝑋) → ((𝐶𝑅𝑥𝐷 = 𝐴) ↔ ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
1110rexbidva 3044 . 2 (𝜑 → (∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴) ↔ ∃𝑥𝑋 ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
126, 11bitr4d 271 1 (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2909  Vcvv 3190  cop 4161   class class class wbr 4623  cmpt 4683  ran crn 5085   Er wer 7699  [cec 7700   / cqs 7701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-er 7702  df-ec 7704  df-qs 7708
This theorem is referenced by: (None)
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