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Theorem qliftfuns 8823
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.) (Revised by AV, 3-Aug-2024.)
Hypotheses
Ref Expression
qlift.1 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
qlift.2 ((𝜑𝑥𝑋) → 𝐴𝑌)
qlift.3 (𝜑𝑅 Er 𝑋)
qlift.4 (𝜑𝑋𝑉)
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)   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem qliftfuns
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 nfcv 2891 . . . . 5 𝑦⟨[𝑥]𝑅, 𝐴
3 nfcv 2891 . . . . . 6 𝑥[𝑦]𝑅
4 nfcsb1v 3914 . . . . . 6 𝑥𝑦 / 𝑥𝐴
53, 4nfop 4891 . . . . 5 𝑥⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴
6 eceq1 8763 . . . . . 6 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
7 csbeq1a 3903 . . . . . 6 (𝑥 = 𝑦𝐴 = 𝑦 / 𝑥𝐴)
86, 7opeq12d 4883 . . . . 5 (𝑥 = 𝑦 → ⟨[𝑥]𝑅, 𝐴⟩ = ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
92, 5, 8cbvmpt 5260 . . . 4 (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
109rneqi 5939 . . 3 ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩) = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
111, 10eqtri 2753 . 2 𝐹 = ran (𝑦𝑋 ↦ ⟨[𝑦]𝑅, 𝑦 / 𝑥𝐴⟩)
12 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
1312ralrimiva 3135 . . 3 (𝜑 → ∀𝑥𝑋 𝐴𝑌)
144nfel1 2908 . . . 4 𝑥𝑦 / 𝑥𝐴𝑌
157eleq1d 2810 . . . 4 (𝑥 = 𝑦 → (𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1614, 15rspc 3594 . . 3 (𝑦𝑋 → (∀𝑥𝑋 𝐴𝑌𝑦 / 𝑥𝐴𝑌))
1713, 16mpan9 505 . 2 ((𝜑𝑦𝑋) → 𝑦 / 𝑥𝐴𝑌)
18 qlift.3 . 2 (𝜑𝑅 Er 𝑋)
19 qlift.4 . 2 (𝜑𝑋𝑉)
20 csbeq1 3892 . 2 (𝑦 = 𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)
2111, 17, 18, 19, 20qliftfun 8821 1 (𝜑 → (Fun 𝐹 ↔ ∀𝑦𝑧(𝑦𝑅𝑧𝑦 / 𝑥𝐴 = 𝑧 / 𝑥𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wcel 2098  wral 3050  csb 3889  cop 4636   class class class wbr 5149  cmpt 5232  ran crn 5679  Fun wfun 6543   Er wer 8722  [cec 8723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-er 8725  df-ec 8727  df-qs 8731
This theorem is referenced by: (None)
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