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Theorem qliftfund 6617
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 115 . . 3 (𝜑 → (𝑥𝑅𝑦𝐴 = 𝐵))
32alrimivv 1875 . 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 6616 . 2 (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝑦(𝑥𝑅𝑦𝐴 = 𝐵)))
103, 9mpbird 167 1 (𝜑 → Fun 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wal 1351   = wceq 1353  wcel 2148  Vcvv 2737  cop 3595   class class class wbr 4003  cmpt 4064  ran crn 4627  Fun wfun 5210   Er wer 6531  [cec 6532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-er 6534  df-ec 6536  df-qs 6540
This theorem is referenced by: (None)
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