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Theorem qliftf 8845
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 8838 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftf 7335 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7 df-qs 8751 . . . . 5 (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
8 eqid 2737 . . . . . 6 (𝑥𝑋 ↦ [𝑥]𝑅) = (𝑥𝑋 ↦ [𝑥]𝑅)
98rnmpt 5968 . . . . 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 6722 . 2 (𝜑 → (𝐹:(𝑋 / 𝑅)⟶𝑌𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
136, 12bitr4d 282 1 (𝜑 → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  cop 4632  cmpt 5225  ran crn 5686  Fun wfun 6555  wf 6557   Er wer 8742  [cec 8743   / cqs 8744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-er 8745  df-ec 8747  df-qs 8751
This theorem is referenced by:  orbsta  19331  frgpupf  19791
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