Theorem qusex 13272

Description: Existence of a quotient structure. (Contributed by Jim Kingdon, 25-Apr-2025.)

Assertion

Ref	Expression
qusex	|- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )

Proof of Theorem qusex

Dummy variables e r x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2788	. . . 4 |- ( R e. V -> R e. _V )
2	1	adantr 276	. . 3 |- ( ( R e. V /\ .~ e. W ) -> R e. _V )
3		elex 2788	. . . 4 |- ( .~ e. W -> .~ e. _V )
4	3	adantl 277	. . 3 |- ( ( R e. V /\ .~ e. W ) -> .~ e. _V )
5		basfn 13005	. . . . . 6 |- Base Fn _V
6		funfvex 5616	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
7	6	funfni 5395	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
8	5, 2, 7	sylancr 414	. . . . 5 |- ( ( R e. V /\ .~ e. W ) -> ( Base ` R ) e. _V )
9	8	mptexd 5834	. . . 4 |- ( ( R e. V /\ .~ e. W ) -> ( x e. ( Base ` R ) |-> [ x ] .~ ) e. _V )
10		simpl 109	. . . 4 |- ( ( R e. V /\ .~ e. W ) -> R e. V )
11		imasex 13252	. . . 4 |- ( ( ( x e. ( Base ` R ) |-> [ x ] .~ ) e. _V /\ R e. V ) -> ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) e. _V )
12	9, 10, 11	syl2anc 411	. . 3 |- ( ( R e. V /\ .~ e. W ) -> ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) e. _V )
13		fveq2 5599	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
14	13	mpteq1d 4145	. . . . 5 |- ( r = R -> ( x e. ( Base ` r ) |-> [ x ] e ) = ( x e. ( Base ` R ) |-> [ x ] e ) )
15		id 19	. . . . 5 |- ( r = R -> r = R )
16	14, 15	oveq12d 5985	. . . 4 |- ( r = R -> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) = ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) )
17		eceq2 6680	. . . . . 6 |- ( e = .~ -> [ x ] e = [ x ] .~ )
18	17	mpteq2dv 4151	. . . . 5 |- ( e = .~ -> ( x e. ( Base ` R ) |-> [ x ] e ) = ( x e. ( Base ` R ) |-> [ x ] .~ ) )
19	18	oveq1d 5982	. . . 4 |- ( e = .~ -> ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
20		df-qus 13250	. . . 4 |- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
21	16, 19, 20	ovmpog 6103	. . 3 |- ( ( R e. _V /\ .~ e. _V /\ ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) e. _V ) -> ( R /.s .~ ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
22	2, 4, 12, 21	syl3anc 1250	. 2 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
23	22, 12	eqeltrd 2284	1 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   |-> cmpt 4121   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   [cec 6641   Basecbs 12947   "_s cimas 13246   /._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-in1 615  ax-in2 616  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-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  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-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  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-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-ec 6645  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-plusg 13037  df-mulr 13038  df-iimas 13249  df-qus 13250

This theorem is referenced by:  znval  14513  znle  14514  znbaslemnn  14516