Theorem quslem 17447

Description: The function in qusval 17446 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusval.u	⊢ (𝜑 → 𝑈 = (𝑅 /s ∼ ))
qusval.v	⊢ (𝜑 → 𝑉 = (Base‘𝑅))
qusval.f	⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
qusval.e	⊢ (𝜑 → ∼ ∈ 𝑊)
qusval.r	⊢ (𝜑 → 𝑅 ∈ 𝑍)

Assertion

Ref	Expression
quslem	⊢ (𝜑 → 𝐹:𝑉–onto→(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.

Step	Hyp	Ref	Expression
1		qusval.e	. . . . . 6 ⊢ (𝜑 → ∼ ∈ 𝑊)
2		ecexg 8626	. . . . . 6 ⊢ ( ∼ ∈ 𝑊 → [𝑥] ∼ ∈ V)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → [𝑥] ∼ ∈ V)
4	3	ralrimivw 3128	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V)
5		qusval.f	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ [𝑥] ∼ )
6	5	fnmpt 6621	. . . 4 ⊢ (∀𝑥 ∈ 𝑉 [𝑥] ∼ ∈ V → 𝐹 Fn 𝑉)
7	4, 6	syl 17	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝑉)
8		dffn4 6741	. . 3 ⊢ (𝐹 Fn 𝑉 ↔ 𝐹:𝑉–onto→ran 𝐹)
9	7, 8	sylib 218	. 2 ⊢ (𝜑 → 𝐹:𝑉–onto→ran 𝐹)
10	5	rnmpt 5896	. . . 4 ⊢ ran 𝐹 = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
11		df-qs 8628	. . . 4 ⊢ (𝑉 / ∼ ) = {𝑦 ∣ ∃𝑥 ∈ 𝑉 𝑦 = [𝑥] ∼ }
12	10, 11	eqtr4i 2757	. . 3 ⊢ ran 𝐹 = (𝑉 / ∼ )
13		foeq3 6733	. . 3 ⊢ (ran 𝐹 = (𝑉 / ∼ ) → (𝐹:𝑉–onto→ran 𝐹 ↔ 𝐹:𝑉–onto→(𝑉 / ∼ )))
14	12, 13	ax-mp 5	. 2 ⊢ (𝐹:𝑉–onto→ran 𝐹 ↔ 𝐹:𝑉–onto→(𝑉 / ∼ ))
15	9, 14	sylib 218	1 ⊢ (𝜑 → 𝐹:𝑉–onto→(𝑉 / ∼ ))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ↦ cmpt 5170  ran crn 5615   Fn wfn 6476  –onto→wfo 6479  ‘cfv 6481  (class class class)co 7346  [cec 8620   / cqs 8621  Basecbs 17120   /_s cqus 17409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-fo 6487  df-ec 8624  df-qs 8628

This theorem is referenced by:  qusbas  17449  quss  17450  qusaddvallem  17455  qusaddflem  17456  qusaddval  17457  qusaddf  17458  qusmulval  17459  qusmulf  17460  qusgrp2  18971  qusrng  20098  qusring2  20252  znzrhfo  21484  qustps  23637  qustgpopn  24035  qustgplem  24036  qustgphaus  24038  qusker  33314  qusvsval  33317  quslmod  33323  quslmhm  33324  qusdimsum  33641