Theorem quslem 13406

Description: The function in qusval 13405 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 6705	. . . . . 6 |- ( .~ e. W -> [ x ] .~ e. _V )
3	1, 2	syl 14	. . . . 5 |- ( ph -> [ x ] .~ e. _V )
4	3	ralrimivw 2606	. . . 4 |- ( ph -> A. x e. V [ x ] .~ e. _V )
5		qusval.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
6	5	fnmpt 5459	. . . 4 |- ( A. x e. V [ x ] .~ e. _V -> F Fn V )
7	4, 6	syl 14	. . 3 |- ( ph -> F Fn V )
8		dffn4 5565	. . 3 |- ( F Fn V <-> F : V -onto-> ran F )
9	7, 8	sylib 122	. 2 |- ( ph -> F : V -onto-> ran F )
10	5	rnmpt 4980	. . . 4 |- ran F = { y | E. x e. V y = [ x ] .~ }
11		df-qs 6707	. . . 4 |- ( V /. .~ ) = { y | E. x e. V y = [ x ] .~ }
12	10, 11	eqtr4i 2255	. . 3 |- ran F = ( V /. .~ )
13		foeq3 5557	. . 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 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   _Vcvv 2802   |-> cmpt 4150   ran crn 4726   Fn wfn 5321   -onto->wfo 5324   ` cfv 5326  (class class class)co 6017   [cec 6699   /.cqs 6700   Basecbs 13081   /._s cqus 13382

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328  df-fn 5329  df-fo 5332  df-ec 6703  df-qs 6707

This theorem is referenced by:  qusbas  13409  qusaddvallemg  13415  qusaddflemg  13416  qusaddval  13417  qusaddf  13418  qusmulval  13419  qusmulf  13420  qusgrp2  13699  qusrng  13970  qusring2  14078  znzrhfo  14661