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Theorem qliftfuns 7779
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 2761 . . . . 5 𝑦⟨[𝑥]𝑅, 𝐴
3 nfcv 2761 . . . . . 6 𝑥[𝑦]𝑅
4 nfcsb1v 3530 . . . . . 6 𝑥𝑦 / 𝑥𝐴
53, 4nfop 4386 . . . . 5 𝑥⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴
6 eceq1 7727 . . . . . 6 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 csbeq1a 3523 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
86, 7opeq12d 4378 . . . . 5 (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
92, 5, 8cbvmpt 4709 . . . 4 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
109rneqi 5312 . . 3 ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
111, 10eqtri 2643 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
12 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
1312ralrimiva 2960 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
144nfel1 2775 . . . 4 𝑥𝑦 / 𝑥𝐴𝑌
157eleq1d 2683 . . . 4 (𝑥 = 𝑦 → (𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1614, 15rspc 3289 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1713, 16mpan9 486 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑌)
18 qlift.3 . 2 (𝜑𝑅 Er 𝑋)
19 qlift.4 . 2 (𝜑𝑋 ∈ V)
20 csbeq1 3517 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2111, 17, 18, 19, 20qliftfun 7777 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  csb 3514  cop 4154   class class class wbr 4613  cmpt 4673  ran crn 5075  Fun wfun 5841   Er wer 7684  [cec 7685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-er 7687  df-ec 7689  df-qs 7693
This theorem is referenced by: (None)
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