Theorem qusin 12800

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 6635	. . . . 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 4108	. . 3 |- ( ph -> ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) )
5	4	oveq1d 5910	. 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 2189	. . 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 12797	. 2 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
12		eqidd 2190	. . 3 |- ( ph -> ( R /.s ( .~ i^i ( V X. V ) ) ) = ( R /.s ( .~ i^i ( V X. V ) ) ) )
13		eqid 2189	. . 3 |- ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) )
14		inex1g 4154	. . . 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 12797	. 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 2232	1 |- ( ph -> U = ( R /.s ( .~ i^i ( V X. V ) ) ) )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2160   _Vcvv 2752   i^i cin 3143   C_ wss 3144   |-> cmpt 4079   X. cxp 4642   "cima 4647   ` cfv 5235  (class class class)co 5895   [cec 6556   Basecbs 12511   "_s cimas 12773   /._s cqus 12774

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-tp 3615  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-ec 6560  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-plusg 12599  df-mulr 12600  df-iimas 12776  df-qus 12777

This theorem is referenced by: (None)