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Theorem qliftval 8831
Description: The value of the function 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)
qliftval.4 (𝑥 = 𝐶𝐴 = 𝐵)
qliftval.6 (𝜑 → Fun 𝐹)
Assertion
Ref Expression
qliftval ((𝜑𝐶𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem qliftval
StepHypRef Expression
1 qlift.1 . 2 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 qlift.2 . . 3 ((𝜑𝑥𝑋) → 𝐴𝑌)
3 qlift.3 . . 3 (𝜑𝑅 Er 𝑋)
4 qlift.4 . . 3 (𝜑𝑋𝑉)
51, 2, 3, 4qliftlem 8823 . 2 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 8769 . 2 (𝑥 = 𝐶 → [𝑥]𝑅 = [𝐶]𝑅)
7 qliftval.4 . 2 (𝑥 = 𝐶𝐴 = 𝐵)
8 qliftval.6 . 2 (𝜑 → Fun 𝐹)
91, 5, 2, 6, 7, 8fliftval 7330 1 ((𝜑𝐶𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cop 4638  cmpt 5235  ran crn 5683  Fun wfun 6547  cfv 6553   Er wer 8728  [cec 8729   / cqs 8730
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fv 6561  df-er 8731  df-ec 8733  df-qs 8737
This theorem is referenced by:  orbstaval  19270  frgpupval  19736
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