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Theorem qliftfund 7876
 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)
qliftfun.4 (𝑥 = 𝑦𝐴 = 𝐵)
qliftfund.6 ((𝜑𝑥𝑅𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
qliftfund (𝜑 → Fun 𝐹)
Distinct variable groups:   𝑦,𝐴   𝑥,𝐵   𝑥,𝑦,𝜑   𝑥,𝑅,𝑦   𝑦,𝐹   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐹(𝑥)

Proof of Theorem qliftfund
StepHypRef Expression
1 qliftfund.6 . . . 4 ((𝜑𝑥𝑅𝑦) → 𝐴 = 𝐵)
21ex 449 . . 3 (𝜑 → (𝑥𝑅𝑦𝐴 = 𝐵))
32alrimivv 1896 . 2 (𝜑 → ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵))
4 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
5 qlift.2 . . 3 ((𝜑𝑥𝑋) → 𝐴𝑌)
6 qlift.3 . . 3 (𝜑𝑅 Er 𝑋)
7 qlift.4 . . 3 (𝜑𝑋 ∈ V)
8 qliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
94, 5, 6, 7, 8qliftfun 7875 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
103, 9mpbird 247 1 (𝜑 → Fun 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1521   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   class class class wbr 4685   ↦ cmpt 4762  ran crn 5144  Fun wfun 5920   Er wer 7784  [cec 7785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-er 7787  df-ec 7789  df-qs 7793 This theorem is referenced by:  orbstafun  17790  frgpupf  18232
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