ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  qliftval GIF version

Theorem qliftval 6223
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 6215 . 2 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
6 eceq1 6172 . 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 5468 1 ((𝜑𝐶𝑋) → (𝐹‘[𝐶]𝑅) = 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  cop 3406  cmpt 3846  ran crn 4374  Fun wfun 4924  cfv 4930   Er wer 6134  [cec 6135   / cqs 6136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fv 4938  df-er 6137  df-ec 6139  df-qs 6143
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator