Theorem qusin 13158

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 6697	. . . . 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 4134	. . 3 |- ( ph -> ( x e. V |-> [ x ] .~ ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) )
5	4	oveq1d 5959	. 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 2205	. . 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 13155	. 2 |- ( ph -> U = ( ( x e. V |-> [ x ] .~ ) "s R ) )
12		eqidd 2206	. . 3 |- ( ph -> ( R /.s ( .~ i^i ( V X. V ) ) ) = ( R /.s ( .~ i^i ( V X. V ) ) ) )
13		eqid 2205	. . 3 |- ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) ) = ( x e. V |-> [ x ] ( .~ i^i ( V X. V ) ) )
14		inex1g 4180	. . . 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 13155	. 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 2248	1 |- ( ph -> U = ( R /.s ( .~ i^i ( V X. V ) ) ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   i^i cin 3165   C_ wss 3166   |-> cmpt 4105   X. cxp 4673   "cima 4678   ` cfv 5271  (class class class)co 5944   [cec 6618   Basecbs 12832   "_s cimas 13131   /._s cqus 13132

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-tp 3641  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-ec 6622  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-mulr 12923  df-iimas 13134  df-qus 13135

This theorem is referenced by: (None)