Theorem quslem 13271

Description: The function in qusval 13270 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 6647	. . . . . 6 |- ( .~ e. W -> [ x ] .~ e. _V )
3	1, 2	syl 14	. . . . 5 |- ( ph -> [ x ] .~ e. _V )
4	3	ralrimivw 2582	. . . 4 |- ( ph -> A. x e. V [ x ] .~ e. _V )
5		qusval.f	. . . . 5 |- F = ( x e. V |-> [ x ] .~ )
6	5	fnmpt 5422	. . . 4 |- ( A. x e. V [ x ] .~ e. _V -> F Fn V )
7	4, 6	syl 14	. . 3 |- ( ph -> F Fn V )
8		dffn4 5526	. . 3 |- ( F Fn V <-> F : V -onto-> ran F )
9	7, 8	sylib 122	. 2 |- ( ph -> F : V -onto-> ran F )
10	5	rnmpt 4945	. . . 4 |- ran F = { y | E. x e. V y = [ x ] .~ }
11		df-qs 6649	. . . 4 |- ( V /. .~ ) = { y | E. x e. V y = [ x ] .~ }
12	10, 11	eqtr4i 2231	. . 3 |- ran F = ( V /. .~ )
13		foeq3 5518	. . 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 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   |-> cmpt 4121   ran crn 4694   Fn wfn 5285   -onto->wfo 5288   ` cfv 5290  (class class class)co 5967   [cec 6641   /.cqs 6642   Basecbs 12947   /._s cqus 13247

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292  df-fn 5293  df-fo 5296  df-ec 6645  df-qs 6649

This theorem is referenced by:  qusbas  13274  qusaddvallemg  13280  qusaddflemg  13281  qusaddval  13282  qusaddf  13283  qusmulval  13284  qusmulf  13285  qusgrp2  13564  qusrng  13835  qusring2  13943  znzrhfo  14525