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Theorem qliftf 7878
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 7871 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftf 6605 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7 df-qs 7793 . . . . 5 (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
8 eqid 2651 . . . . . 6 (𝑥𝑋 ↦ [𝑥]𝑅) = (𝑥𝑋 ↦ [𝑥]𝑅)
98rnmpt 5403 . . . . 5 ran (𝑥𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
107, 9eqtr4i 2676 . . . 4 (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅)
1110a1i 11 . . 3 (𝜑 → (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅))
1211feq2d 6069 . 2 (𝜑 → (𝐹:(𝑋 / 𝑅)⟶𝑌𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
136, 12bitr4d 271 1 (𝜑 → (Fun 𝐹𝐹:(𝑋 / 𝑅)⟶𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231  cop 4216  cmpt 4762  ran crn 5144  Fun wfun 5920  wf 5922   Er wer 7784  [cec 7785   / cqs 7786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-er 7787  df-ec 7789  df-qs 7793
This theorem is referenced by:  orbsta  17792  frgpupf  18232
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