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Theorem qliftfuns 8072
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 2941 . . . . 5 𝑦⟨[𝑥]𝑅, 𝐴
3 nfcv 2941 . . . . . 6 𝑥[𝑦]𝑅
4 nfcsb1v 3744 . . . . . 6 𝑥𝑦 / 𝑥𝐴
53, 4nfop 4609 . . . . 5 𝑥⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴
6 eceq1 8020 . . . . . 6 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 csbeq1a 3737 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
86, 7opeq12d 4601 . . . . 5 (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
92, 5, 8cbvmpt 4942 . . . 4 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
109rneqi 5555 . . 3 ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
111, 10eqtri 2821 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
12 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
1312ralrimiva 3147 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
144nfel1 2956 . . . 4 𝑥𝑦 / 𝑥𝐴𝑌
157eleq1d 2863 . . . 4 (𝑥 = 𝑦 → (𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1614, 15rspc 3491 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1713, 16mpan9 503 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑌)
18 qlift.3 . 2 (𝜑𝑅 Er 𝑋)
19 qlift.4 . 2 (𝜑𝑋 ∈ V)
20 csbeq1 3731 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2111, 17, 18, 19, 20qliftfun 8070 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  csb 3728  cop 4374   class class class wbr 4843  cmpt 4922  ran crn 5313  Fun wfun 6095   Er wer 7979  [cec 7980
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-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  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-fn 6104  df-f 6105  df-fv 6109  df-er 7982  df-ec 7984  df-qs 7988
This theorem is referenced by: (None)
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