Theorem qusex 13646

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 2833	. . . 4 |- ( R e. V -> R e. _V )
2	1	adantr 276	. . 3 |- ( ( R e. V /\ .~ e. W ) -> R e. _V )
3		elex 2833	. . . 4 |- ( .~ e. W -> .~ e. _V )
4	3	adantl 277	. . 3 |- ( ( R e. V /\ .~ e. W ) -> .~ e. _V )
5		basfn 13411	. . . . . 6 |- Base Fn _V
6		funfvex 5712	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
7	6	funfni 5483	. . . . . 6 |- ( ( Base Fn _V /\ R e. _V ) -> ( Base ` R ) e. _V )
8	5, 2, 7	sylancr 418	. . . . 5 |- ( ( R e. V /\ .~ e. W ) -> ( Base ` R ) e. _V )
9	8	mptexd 5944	. . . 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 13626	. . . 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 415	. . 3 |- ( ( R e. V /\ .~ e. W ) -> ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) e. _V )
13		fveq2 5695	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
14	13	mpteq1d 4216	. . . . 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 6103	. . . 4 |- ( r = R -> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) = ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) )
17		eceq2 6844	. . . . . 6 |- ( e = .~ -> [ x ] e = [ x ] .~ )
18	17	mpteq2dv 4222	. . . . 5 |- ( e = .~ -> ( x e. ( Base ` R ) |-> [ x ] e ) = ( x e. ( Base ` R ) |-> [ x ] .~ ) )
19	18	oveq1d 6100	. . . 4 |- ( e = .~ -> ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
20		df-qus 13625	. . . 4 |- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
21	16, 19, 20	ovmpog 6223	. . 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 1278	. 2 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
23	22, 12	eqeltrd 2315	1 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   |-> cmpt 4192   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   [cec 6805   Basecbs 13352   "_s cimas 13622   /._s cqus 13623

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-tp 3717  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-ec 6809  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-plusg 13444  df-mulr 13445  df-iimas 13624  df-qus 13625

This theorem is used by:  znval  14971  znle  14972  znbaslemnn  14974