Theorem qusin 13408

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 6778	. . . . 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 4179	. . 3 |- ( ph -> ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) )
5	4	oveq1d 6032	. 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 2231	. . 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 13405	. 2 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
12		eqidd 2232	. . 3 |- ( ph -> ( R /.s ( .~ i^i ( V X. V ) ) ) = ( R /.s ( .~ i^i ( V X. V ) ) ) )
13		eqid 2231	. . 3 |- ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) )
14		inex1g 4225	. . . 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 13405	. 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 2274	1 |- ( ph -> U = ( R /.s ( .~ i^i ( V X. V ) ) ) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   _Vcvv 2802   i^i cin 3199   C_ wss 3200   |-> cmpt 4150   X. cxp 4723   "cima 4728   ` cfv 5326  (class class class)co 6017   [cec 6699   Basecbs 13081   "_s cimas 13381   /._s cqus 13382

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-tp 3677  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-ec 6703  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-plusg 13172  df-mulr 13173  df-iimas 13384  df-qus 13385

This theorem is referenced by: (None)