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Theorem quslem 17590
Description: The function in qusval 17589 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
qusval.u (𝜑𝑈 = (𝑅 /s ))
qusval.v (𝜑𝑉 = (Base‘𝑅))
qusval.f 𝐹 = (𝑥𝑉 ↦ [𝑥] )
qusval.e (𝜑𝑊)
qusval.r (𝜑𝑅𝑍)
Assertion
Ref Expression
quslem (𝜑𝐹:𝑉onto→(𝑉 / ))
Distinct variable groups:   𝑥,   𝜑,𝑥   𝑥,𝑅   𝑥,𝑉
Allowed substitution hints:   𝑈(𝑥)   𝐹(𝑥)   𝑊(𝑥)   𝑍(𝑥)

Proof of Theorem quslem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 qusval.e . . . . . 6 (𝜑𝑊)
2 ecexg 8748 . . . . . 6 ( 𝑊 → [𝑥] ∈ V)
31, 2syl 17 . . . . 5 (𝜑 → [𝑥] ∈ V)
43ralrimivw 3148 . . . 4 (𝜑 → ∀𝑥𝑉 [𝑥] ∈ V)
5 qusval.f . . . . 5 𝐹 = (𝑥𝑉 ↦ [𝑥] )
65fnmpt 6709 . . . 4 (∀𝑥𝑉 [𝑥] ∈ V → 𝐹 Fn 𝑉)
74, 6syl 17 . . 3 (𝜑𝐹 Fn 𝑉)
8 dffn4 6827 . . 3 (𝐹 Fn 𝑉𝐹:𝑉onto→ran 𝐹)
97, 8sylib 218 . 2 (𝜑𝐹:𝑉onto→ran 𝐹)
105rnmpt 5971 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
11 df-qs 8750 . . . 4 (𝑉 / ) = {𝑦 ∣ ∃𝑥𝑉 𝑦 = [𝑥] }
1210, 11eqtr4i 2766 . . 3 ran 𝐹 = (𝑉 / )
13 foeq3 6819 . . 3 (ran 𝐹 = (𝑉 / ) → (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / )))
1412, 13ax-mp 5 . 2 (𝐹:𝑉onto→ran 𝐹𝐹:𝑉onto→(𝑉 / ))
159, 14sylib 218 1 (𝜑𝐹:𝑉onto→(𝑉 / ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cmpt 5231  ran crn 5690   Fn wfn 6558  ontowfo 6561  cfv 6563  (class class class)co 7431  [cec 8742   / cqs 8743  Basecbs 17245   /s cqus 17552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-fo 6569  df-ec 8746  df-qs 8750
This theorem is referenced by:  qusbas  17592  quss  17593  qusaddvallem  17598  qusaddflem  17599  qusaddval  17600  qusaddf  17601  qusmulval  17602  qusmulf  17603  qusgrp2  19089  qusrng  20198  qusring2  20348  znzrhfo  21584  qustps  23746  qustgpopn  24144  qustgplem  24145  qustgphaus  24147  qusker  33357  qusvsval  33360  quslmod  33366  quslmhm  33367  qusdimsum  33656
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