Theorem qusin 12746

Description: Restrict the equivalence relation in a quotient structure to the base set. (Contributed by Mario Carneiro, 23-Feb-2015.)

Hypotheses

Ref	Expression
qusin.u	|- ( ph -> U = ( R /.s .~ ) )
qusin.v	|- ( ph -> V = ( Base ` R ) )
qusin.e	|- ( ph -> .~ e. W )
qusin.r	|- ( ph -> R e. Z )
qusin.s	|- ( ph -> ( .~ " V ) C_ V )

Assertion

Ref	Expression
qusin	|- ( ph -> U = ( R /.s ( .~ i^i ( V X. V ) ) ) )

Proof of Theorem qusin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		qusin.s	. . . . 5 |- ( ph -> ( .~ " V ) C_ V )
2		ecinxp 6610	. . . . 5 |- ( ( ( .~ " V ) C_ V /\ x e. V ) -> [ x ] .~ = [ x ] ( .~ i^i ( V X. V ) ) )
3	1, 2	sylan 283	. . . 4 |- ( ( ph /\ x e. V ) -> [ x ] .~ = [ x ] ( .~ i^i ( V X. V ) ) )
4	3	mpteq2dva 4094	. . 3 |- ( ph -> ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) )
5	4	oveq1d 5890	. 2 |- ( ph -> ( ( x e. V |-> [ x ] .~ ) "s R ) = ( ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) "s R ) )
6		qusin.u	. . 3 |- ( ph -> U = ( R /.s .~ ) )
7		qusin.v	. . 3 |- ( ph -> V = ( Base ` R ) )
8		eqid 2177	. . 3 |- ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] .~ )
9		qusin.e	. . 3 |- ( ph -> .~ e. W )
10		qusin.r	. . 3 |- ( ph -> R e. Z )
11	6, 7, 8, 9, 10	qusval 12744	. 2 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
12		eqidd 2178	. . 3 |- ( ph -> ( R /.s ( .~ i^i ( V X. V ) ) ) = ( R /.s ( .~ i^i ( V X. V ) ) ) )
13		eqid 2177	. . 3 |- ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) )
14		inex1g 4140	. . . 4 |- ( .~ e. W -> ( .~ i^i ( V X. V ) ) e. _V )
15	9, 14	syl 14	. . 3 |- ( ph -> ( .~ i^i ( V X. V ) ) e. _V )
16	12, 7, 13, 15, 10	qusval 12744	. 2 |- ( ph -> ( R /.s ( .~ i^i ( V X. V ) ) ) = ( ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) "s R ) )
17	5, 11, 16	3eqtr4d 2220	1 |- ( ph -> U = ( R /.s ( .~ i^i ( V X. V ) ) ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2738   i^i cin 3129   C_ wss 3130   |-> cmpt 4065   X. cxp 4625   "cima 4630   ` cfv 5217  (class class class)co 5875   [cec 6533   Basecbs 12462   "_s cimas 12720   /._s cqus 12721

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-cnex 7902  ax-resscn 7903  ax-1re 7905  ax-addrcl 7908

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-tp 3601  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-ec 6537  df-inn 8920  df-2 8978  df-3 8979  df-ndx 12465  df-slot 12466  df-base 12468  df-plusg 12549  df-mulr 12550  df-iimas 12723  df-qus 12724

This theorem is referenced by: (None)