Theorem qusex 13488

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 2815	. . . 4 |- ( R e. V -> R e. _V )
2	1	adantr 276	. . 3 |- ( ( R e. V /\ .~ e. W ) -> R e. _V )
3		elex 2815	. . . 4 |- ( .~ e. W -> .~ e. _V )
4	3	adantl 277	. . 3 |- ( ( R e. V /\ .~ e. W ) -> .~ e. _V )
5		basfn 13221	. . . . . 6 |- Base Fn _V
6		funfvex 5665	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
7	6	funfni 5439	. . . . . 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 5891	. . . 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 13468	. . . 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 5648	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
14	13	mpteq1d 4179	. . . . 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 6046	. . . 4 |- ( r = R -> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) = ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) )
17		eceq2 6782	. . . . . 6 |- ( e = .~ -> [ x ] e = [ x ] .~ )
18	17	mpteq2dv 4185	. . . . 5 |- ( e = .~ -> ( x e. ( Base ` R ) |-> [ x ] e ) = ( x e. ( Base ` R ) |-> [ x ] .~ ) )
19	18	oveq1d 6043	. . . 4 |- ( e = .~ -> ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
20		df-qus 13466	. . . 4 |- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
21	16, 19, 20	ovmpog 6166	. . 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 1274	. 2 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
23	22, 12	eqeltrd 2308	1 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   |-> cmpt 4155   Fn wfn 5328   ` cfv 5333  (class class class)co 6028   [cec 6743   Basecbs 13162   "_s cimas 13462   /._s cqus 13463

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-tp 3681  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-ec 6747  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-plusg 13253  df-mulr 13254  df-iimas 13465  df-qus 13466

This theorem is referenced by:  znval  14732  znle  14733  znbaslemnn  14735