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Theorem qliftfund 8743
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)
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)   𝑉(𝑥,𝑦)

Proof of Theorem qliftfund
StepHypRef Expression
1 qliftfund.6 . . . 4 ((𝜑𝑥𝑅𝑦) → 𝐴 = 𝐵)
21ex 414 . . 3 (𝜑 → (𝑥𝑅𝑦𝐴 = 𝐵))
32alrimivv 1932 . 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 (𝜑𝑋𝑉)
8 qliftfun.4 . . 3 (𝑥 = 𝑦𝐴 = 𝐵)
94, 5, 6, 7, 8qliftfun 8742 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
103, 9mpbird 257 1 (𝜑 → Fun 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  cop 4593   class class class wbr 5106  cmpt 5189  ran crn 5635  Fun wfun 6491   Er wer 8646  [cec 8647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-er 8649  df-ec 8651  df-qs 8655
This theorem is referenced by:  orbstafun  19092  frgpupf  19556
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