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Theorem qliftf 8702
Description: The domain and codomain of the function 𝐹. (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
qliftf (𝜑 → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
Distinct variable groups:   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem qliftf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 qlift.1 . . 3 𝐹 = ran (𝑥𝑋 ↦ ⟨[𝑥]𝑅, 𝐴⟩)
2 qlift.2 . . . 4 ((𝜑𝑥𝑋) → 𝐴𝑌)
3 qlift.3 . . . 4 (𝜑𝑅 Er 𝑋)
4 qlift.4 . . . 4 (𝜑𝑋𝑉)
51, 2, 3, 4qliftlem 8695 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftf 7256 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7 df-qs 8612 . . . . 5 (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
8 eqid 2737 . . . . . 6 (𝑥𝑋 ↦ [𝑥]𝑅) = (𝑥𝑋 ↦ [𝑥]𝑅)
98rnmpt 5908 . . . . 5 ran (𝑥𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
107, 9eqtr4i 2768 . . . 4 (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅)
1110a1i 11 . . 3 (𝜑 → (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅))
1211feq2d 6651 . 2 (𝜑 → (𝐹:(𝑋 / 𝑅)⟶𝑌𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
136, 12bitr4d 281 1 (𝜑 → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2714  wrex 3071  cop 4590  cmpt 5186  ran crn 5632  Fun wfun 6487  wf 6489   Er wer 8603  [cec 8604   / cqs 8605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6495  df-fn 6496  df-f 6497  df-er 8606  df-ec 8608  df-qs 8612
This theorem is referenced by:  orbsta  19052  frgpupf  19514
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