Theorem qusex 12968

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 2774	. . . 4 |- ( R e. V -> R e. _V )
2	1	adantr 276	. . 3 |- ( ( R e. V /\ .~ e. W ) -> R e. _V )
3		elex 2774	. . . 4 |- ( .~ e. W -> .~ e. _V )
4	3	adantl 277	. . 3 |- ( ( R e. V /\ .~ e. W ) -> .~ e. _V )
5		basfn 12736	. . . . . 6 |- Base Fn _V
6		funfvex 5575	. . . . . . 7 |- ( ( Fun Base /\ R e. dom Base ) -> ( Base ` R ) e. _V )
7	6	funfni 5358	. . . . . 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 5789	. . . 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 12948	. . . 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 5558	. . . . . 6 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
14	13	mpteq1d 4118	. . . . 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 5940	. . . 4 |- ( r = R -> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) = ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) )
17		eceq2 6629	. . . . . 6 |- ( e = .~ -> [ x ] e = [ x ] .~ )
18	17	mpteq2dv 4124	. . . . 5 |- ( e = .~ -> ( x e. ( Base ` R ) |-> [ x ] e ) = ( x e. ( Base ` R ) |-> [ x ] .~ ) )
19	18	oveq1d 5937	. . . 4 |- ( e = .~ -> ( ( x e. ( Base ` R ) |-> [ x ] e ) "s R ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
20		df-qus 12946	. . . 4 |- /.s = ( r e. _V , e e. _V |-> ( ( x e. ( Base ` r ) |-> [ x ] e ) "s r ) )
21	16, 19, 20	ovmpog 6057	. . 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 1249	. 2 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) = ( ( x e. ( Base ` R ) |-> [ x ] .~ ) "s R ) )
23	22, 12	eqeltrd 2273	1 |- ( ( R e. V /\ .~ e. W ) -> ( R /.s .~ ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   _Vcvv 2763   |-> cmpt 4094   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   [cec 6590   Basecbs 12678   "_s cimas 12942   /._s cqus 12943

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1re 7973  ax-addrcl 7976

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-tp 3630  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-ov 5925  df-oprab 5926  df-mpo 5927  df-ec 6594  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-plusg 12768  df-mulr 12769  df-iimas 12945  df-qus 12946

This theorem is referenced by:  znval  14192  znle  14193  znbaslemnn  14195