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Theorem quslem 17489
Description: The function in qusval 17488 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
qusval.u (πœ‘ β†’ π‘ˆ = (𝑅 /s ∼ ))
qusval.v (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))
qusval.f 𝐹 = (π‘₯ ∈ 𝑉 ↦ [π‘₯] ∼ )
qusval.e (πœ‘ β†’ ∼ ∈ π‘Š)
qusval.r (πœ‘ β†’ 𝑅 ∈ 𝑍)
Assertion
Ref Expression
quslem (πœ‘ β†’ 𝐹:𝑉–ontoβ†’(𝑉 / ∼ ))
Distinct variable groups:   π‘₯, ∼   πœ‘,π‘₯   π‘₯,𝑅   π‘₯,𝑉
Allowed substitution hints:   π‘ˆ(π‘₯)   𝐹(π‘₯)   π‘Š(π‘₯)   𝑍(π‘₯)

Proof of Theorem quslem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 qusval.e . . . . . 6 (πœ‘ β†’ ∼ ∈ π‘Š)
2 ecexg 8707 . . . . . 6 ( ∼ ∈ π‘Š β†’ [π‘₯] ∼ ∈ V)
31, 2syl 17 . . . . 5 (πœ‘ β†’ [π‘₯] ∼ ∈ V)
43ralrimivw 3151 . . . 4 (πœ‘ β†’ βˆ€π‘₯ ∈ 𝑉 [π‘₯] ∼ ∈ V)
5 qusval.f . . . . 5 𝐹 = (π‘₯ ∈ 𝑉 ↦ [π‘₯] ∼ )
65fnmpt 6691 . . . 4 (βˆ€π‘₯ ∈ 𝑉 [π‘₯] ∼ ∈ V β†’ 𝐹 Fn 𝑉)
74, 6syl 17 . . 3 (πœ‘ β†’ 𝐹 Fn 𝑉)
8 dffn4 6812 . . 3 (𝐹 Fn 𝑉 ↔ 𝐹:𝑉–ontoβ†’ran 𝐹)
97, 8sylib 217 . 2 (πœ‘ β†’ 𝐹:𝑉–ontoβ†’ran 𝐹)
105rnmpt 5955 . . . 4 ran 𝐹 = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑉 𝑦 = [π‘₯] ∼ }
11 df-qs 8709 . . . 4 (𝑉 / ∼ ) = {𝑦 ∣ βˆƒπ‘₯ ∈ 𝑉 𝑦 = [π‘₯] ∼ }
1210, 11eqtr4i 2764 . . 3 ran 𝐹 = (𝑉 / ∼ )
13 foeq3 6804 . . 3 (ran 𝐹 = (𝑉 / ∼ ) β†’ (𝐹:𝑉–ontoβ†’ran 𝐹 ↔ 𝐹:𝑉–ontoβ†’(𝑉 / ∼ )))
1412, 13ax-mp 5 . 2 (𝐹:𝑉–ontoβ†’ran 𝐹 ↔ 𝐹:𝑉–ontoβ†’(𝑉 / ∼ ))
159, 14sylib 217 1 (πœ‘ β†’ 𝐹:𝑉–ontoβ†’(𝑉 / ∼ ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  βˆƒwrex 3071  Vcvv 3475   ↦ cmpt 5232  ran crn 5678   Fn wfn 6539  β€“ontoβ†’wfo 6542  β€˜cfv 6544  (class class class)co 7409  [cec 8701   / cqs 8702  Basecbs 17144   /s cqus 17451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-fo 6550  df-ec 8705  df-qs 8709
This theorem is referenced by:  qusbas  17491  quss  17492  qusaddvallem  17497  qusaddflem  17498  qusaddval  17499  qusaddf  17500  qusmulval  17501  qusmulf  17502  qusgrp2  18941  qusring2  20147  znzrhfo  21103  qustps  23226  qustgpopn  23624  qustgplem  23625  qustgphaus  23627  qusker  32464  qusvsval  32467  quslmod  32469  quslmhm  32470  qusdimsum  32713  qusrng  46681
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