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Theorem qliftval 8074
Description: The value of the function 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)
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)

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 (𝜑𝑋 ∈ V)
51, 2, 3, 4qliftlem 8066 . 2 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 8020 . 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 6794 1 ((𝜑𝐶𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  cop 4374  cmpt 4922  ran crn 5313  Fun wfun 6095  cfv 6101   Er wer 7979  [cec 7980   / cqs 7981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-er 7982  df-ec 7984  df-qs 7988
This theorem is referenced by:  orbstaval  18057  frgpupval  18502
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