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Theorem qliftf 7722
 Description: The domain and range of the function 𝐹. (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)
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)

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 (𝜑𝑋 ∈ V)
51, 2, 3, 4qliftlem 7715 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftf 6465 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7 df-qs 7635 . . . . 5 (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
8 eqid 2610 . . . . . 6 (𝑥𝑋 ↦ [𝑥]𝑅) = (𝑥𝑋 ↦ [𝑥]𝑅)
98rnmpt 5292 . . . . 5 ran (𝑥𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
107, 9eqtr4i 2635 . . . 4 (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅)
1110a1i 11 . . 3 (𝜑 → (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅))
1211feq2d 5944 . 2 (𝜑 → (𝐹:(𝑋 / 𝑅)⟶𝑌𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
136, 12bitr4d 270 1 (𝜑 → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∃wrex 2897  Vcvv 3173  ⟨cop 4131   ↦ cmpt 4643  ran crn 5039  Fun wfun 5798  ⟶wf 5800   Er wer 7626  [cec 7627   / cqs 7628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-er 7629  df-ec 7631  df-qs 7635 This theorem is referenced by:  orbsta  17569  frgpupf  18009
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