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Theorem qliftval 6128
 Description: The value of the function 𝐹. (Contributed by Mario Carneiro, 23-Dec-2016.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (x 𝑋 ↦ ⟨[x]𝑅, A⟩)
qlift.2 ((φ x 𝑋) → A 𝑌)
qlift.3 (φ𝑅 Er 𝑋)
qlift.4 (φ𝑋 V)
qliftval.4 (x = 𝐶A = B)
qliftval.6 (φ → Fun 𝐹)
Assertion
Ref Expression
qliftval ((φ 𝐶 𝑋) → (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 (x 𝑋 ↦ ⟨[x]𝑅, A⟩)
2 qlift.2 . . 3 ((φ x 𝑋) → A 𝑌)
3 qlift.3 . . 3 (φ𝑅 Er 𝑋)
4 qlift.4 . . 3 (φ𝑋 V)
51, 2, 3, 4qliftlem 6120 . 2 ((φ x 𝑋) → [x]𝑅 (𝑋 / 𝑅))
6 eceq1 6077 . 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 5383 1 ((φ 𝐶 𝑋) → (𝐹‘[𝐶]𝑅) = B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242   ∈ wcel 1390  Vcvv 2551  ⟨cop 3370   ↦ cmpt 3809  ran crn 4289  Fun wfun 4839  ‘cfv 4845   Er wer 6039  [cec 6040   / cqs 6041 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853  df-er 6042  df-ec 6044  df-qs 6048 This theorem is referenced by: (None)
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