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Theorem qliftel 8793
Description: Elementhood in the relation 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.) (Revised by AV, 3-Aug-2024.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2 ((𝜑𝑥𝑋) → 𝐴𝑌)
qlift.3 (𝜑𝑅 Er 𝑋)
qlift.4 (𝜑𝑋𝑉)
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)   𝑉(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 (𝜑𝑋𝑉)
51, 2, 3, 4qliftlem 8791 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftel 7301 . 2 (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
73adantr 480 . . . . 5 ((𝜑𝑥𝑋) → 𝑅 Er 𝑋)
8 simpr 484 . . . . 5 ((𝜑𝑥𝑋) → 𝑥𝑋)
97, 8erth2 8752 . . . 4 ((𝜑𝑥𝑋) → (𝐶𝑅𝑥 ↔ [𝐶]𝑅 = [𝑥]𝑅))
109anbi1d 629 . . 3 ((𝜑𝑥𝑋) → ((𝐶𝑅𝑥𝐷 = 𝐴) ↔ ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
1110rexbidva 3170 . 2 (𝜑 → (∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴) ↔ ∃𝑥𝑋 ([𝐶]𝑅 = [𝑥]𝑅𝐷 = 𝐴)))
126, 11bitr4d 282 1 (𝜑 → ([𝐶]𝑅𝐹𝐷 ↔ ∃𝑥𝑋 (𝐶𝑅𝑥𝐷 = 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3064  cop 4629   class class class wbr 5141  cmpt 5224  ran crn 5670   Er wer 8699  [cec 8700   / cqs 8701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-er 8702  df-ec 8704  df-qs 8708
This theorem is referenced by: (None)
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