Theorem quslem 12907

Description: The function in qusval 12906 is a surjection onto a quotient set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusval.u	|- ( ph -> U = ( R /.s .~ ) )
qusval.v	|- ( ph -> V = ( Base ` R ) )
qusval.f	|- F = ( x e. V |-> [ x ] .~ )
qusval.e	|- ( ph -> .~ e. W )
qusval.r	|- ( ph -> R e. Z )

Assertion

Ref	Expression
quslem	|- ( ph -> F : V -onto-> ( V /. .~ ) )

Distinct variable groups:   x, .~   ph, 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 |- ( ph -> .~ e. W )
2		ecexg 6591	. . . . . 6 |- ( .~ e. W -> [ x ] .~ e. _V )
3	1, 2	syl 14	. . . . 5 |- ( ph -> [ x ] .~ e. _V )
4	3	ralrimivw 2568	. . . 4 |- ( ph -> A. x e. V [ x ] .~ e. _V )
5		qusval.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
6	5	fnmpt 5380	. . . 4 |- ( A. x e. V [ x ] .~ e. _V -> F Fn V )
7	4, 6	syl 14	. . 3 |- ( ph -> F Fn V )
8		dffn4 5482	. . 3 |- ( F Fn V <-> F : V -onto-> ran F )
9	7, 8	sylib 122	. 2 |- ( ph -> F : V -onto-> ran F )
10	5	rnmpt 4910	. . . 4 |- ran F = { y | E. x e. V y = [ x ] .~ }
11		df-qs 6593	. . . 4 |- ( V /. .~ ) = { y | E. x e. V y = [ x ] .~ }
12	10, 11	eqtr4i 2217	. . 3 |- ran F = ( V /. .~ )
13		foeq3 5474	. . 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 122	1 |- ( ph -> F : V -onto-> ( V /. .~ ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   |-> cmpt 4090   ran crn 4660   Fn wfn 5249   -onto->wfo 5252   ` cfv 5254  (class class class)co 5918   [cec 6585   /.cqs 6586   Basecbs 12618   /._s cqus 12883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-fun 5256  df-fn 5257  df-fo 5260  df-ec 6589  df-qs 6593

This theorem is referenced by:  qusbas  12910  qusaddvallemg  12916  qusaddflemg  12917  qusaddval  12918  qusaddf  12919  qusmulval  12920  qusmulf  12921  qusgrp2  13183  qusrng  13454  qusring2  13562  znzrhfo  14136