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Theorem qliftfuns 6221
 Description: The function 𝐹 is the unique function defined by 𝐹‘[𝑥] = 𝐴, provided that the well-definedness condition holds. (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)
Assertion
Ref Expression
qliftfuns (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
Distinct variable groups:   𝑦,𝑧,𝐴   𝑥,𝑦,𝑧,𝜑   𝑥,𝑅,𝑦,𝑧   𝑦,𝐹,𝑧   𝑥,𝑋,𝑦,𝑧   𝑥,𝑌,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem qliftfuns
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 nfcv 2194 . . . . 5 𝑦⟨[𝑥]𝑅, 𝐴
3 nfcv 2194 . . . . . 6 𝑥[𝑦]𝑅
4 nfcsb1v 2910 . . . . . 6 𝑥𝑦 / 𝑥𝐴
53, 4nfop 3593 . . . . 5 𝑥⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴
6 eceq1 6172 . . . . . 6 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 csbeq1a 2888 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
86, 7opeq12d 3585 . . . . 5 (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
92, 5, 8cbvmpt 3879 . . . 4 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
109rneqi 4590 . . 3 ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
111, 10eqtri 2076 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
12 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
1312ralrimiva 2409 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
144nfel1 2204 . . . 4 𝑥𝑦 / 𝑥𝐴𝑌
157eleq1d 2122 . . . 4 (𝑥 = 𝑦 → (𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1614, 15rspc 2667 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1713, 16mpan9 269 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑌)
18 qlift.3 . 2 (𝜑𝑅 Er 𝑋)
19 qlift.4 . 2 (𝜑𝑋 ∈ V)
20 csbeq1 2883 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2111, 17, 18, 19, 20qliftfun 6219 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102  ∀wal 1257   = wceq 1259   ∈ wcel 1409  ∀wral 2323  Vcvv 2574  ⦋csb 2880  ⟨cop 3406   class class class wbr 3792   ↦ cmpt 3846  ran crn 4374  Fun wfun 4924   Er wer 6134  [cec 6135 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-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  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-fn 4933  df-f 4934  df-fv 4938  df-er 6137  df-ec 6139  df-qs 6143 This theorem is referenced by: (None)
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