Theorem qusin 13359

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 6757	. . . . 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 4174	. . 3 |- ( ph -> ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) )
5	4	oveq1d 6016	. 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 2229	. . 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 13356	. 2 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
12		eqidd 2230	. . 3 |- ( ph -> ( R /.s ( .~ i^i ( V X. V ) ) ) = ( R /.s ( .~ i^i ( V X. V ) ) ) )
13		eqid 2229	. . 3 |- ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) )
14		inex1g 4220	. . . 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 13356	. 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 2272	1 |- ( ph -> U = ( R /.s ( .~ i^i ( V X. V ) ) ) )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   C_ wss 3197   |-> cmpt 4145   X. cxp 4717   "cima 4722   ` cfv 5318  (class class class)co 6001   [cec 6678   Basecbs 13032   "_s cimas 13332   /._s cqus 13333

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1re 8093  ax-addrcl 8096

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-tp 3674  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-ec 6682  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-plusg 13123  df-mulr 13124  df-iimas 13335  df-qus 13336

This theorem is referenced by: (None)