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Theorem qliftf 7987
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 7980 . . 3 ((𝜑𝑥𝑋) → [𝑥]𝑅 ∈ (𝑋 / 𝑅))
61, 5, 2fliftf 6708 . 2 (𝜑 → (Fun 𝐹𝐹:ran (𝑥𝑋 ↦ [𝑥]𝑅)⟶𝑌))
7 df-qs 7902 . . . . 5 (𝑋 / 𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
8 eqid 2771 . . . . . 6 (𝑥𝑋 ↦ [𝑥]𝑅) = (𝑥𝑋 ↦ [𝑥]𝑅)
98rnmpt 5509 . . . . 5 ran (𝑥𝑋 ↦ [𝑥]𝑅) = {𝑦 ∣ ∃𝑥𝑋 𝑦 = [𝑥]𝑅}
107, 9eqtr4i 2796 . . . 4 (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅)
1110a1i 11 . . 3 (𝜑 → (𝑋 / 𝑅) = ran (𝑥𝑋 ↦ [𝑥]𝑅))
1211feq2d 6171 . 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 382   = wceq 1631  wcel 2145  {cab 2757  wrex 3062  Vcvv 3351  cop 4322  cmpt 4863  ran crn 5250  Fun wfun 6025  wf 6027   Er wer 7893  [cec 7894   / cqs 7895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-er 7896  df-ec 7898  df-qs 7902
This theorem is referenced by:  orbsta  17953  frgpupf  18393
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