Theorem inxprnres 34060

Description: Restriction of a class as a class of ordered pairs. (Contributed by Peter Mazsa, 2-Jan-2019.)

Assertion

Ref	Expression
inxprnres	|- ( R i^i ( A X. ran ( R |` A ) ) ) = { <. x , y >. | ( x e. A /\ x R y ) }

Distinct variable groups:   x, A, y   x, R, y

Proof of Theorem inxprnres

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5227	. . 3 |- Rel ( A X. ran ( R |` A ) )
2		relin2 5237	. . 3 |- ( Rel ( A X. ran ( R |` A ) ) -> Rel ( R i^i ( A X. ran ( R |` A ) ) ) )
3	1, 2	ax-mp 5	. 2 |- Rel ( R i^i ( A X. ran ( R |` A ) ) )
4		relopab 5247	. 2 |- Rel { <. x , y >. | ( x e. A /\ x R y ) }
5		eleq1w 2684	. . . . . 6 |- ( x = z -> ( x e. A <-> z e. A ) )
6		breq1 4656	. . . . . 6 |- ( x = z -> ( x R y <-> z R y ) )
7	5, 6	anbi12d 747	. . . . 5 |- ( x = z -> ( ( x e. A /\ x R y ) <-> ( z e. A /\ z R y ) ) )
8		breq2 4657	. . . . . 6 |- ( y = w -> ( z R y <-> z R w ) )
9	8	anbi2d 740	. . . . 5 |- ( y = w -> ( ( z e. A /\ z R y ) <-> ( z e. A /\ z R w ) ) )
10	7, 9	opelopabg 4993	. . . 4 |- ( ( z e. _V /\ w e. _V ) -> ( <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } <-> ( z e. A /\ z R w ) ) )
11	10	el2v 33984	. . 3 |- ( <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } <-> ( z e. A /\ z R w ) )
12		brinxprnres 34059	. . . 4 |- ( w e. _V -> ( z ( R i^i ( A X. ran ( R |` A ) ) ) w <-> ( z e. A /\ z R w ) ) )
13	12	elv 33983	. . 3 |- ( z ( R i^i ( A X. ran ( R |` A ) ) ) w <-> ( z e. A /\ z R w ) )
14		df-br 4654	. . 3 |- ( z ( R i^i ( A X. ran ( R |` A ) ) ) w <-> <. z , w >. e. ( R i^i ( A X. ran ( R |` A ) ) ) )
15	11, 13, 14	3bitr2ri 289	. 2 |- ( <. z , w >. e. ( R i^i ( A X. ran ( R |` A ) ) ) <-> <. z , w >. e. { <. x , y >. | ( x e. A /\ x R y ) } )
16	3, 4, 15	eqrelriiv 5214	1 |- ( R i^i ( A X. ran ( R |` A ) ) ) = { <. x , y >. | ( x e. A /\ x R y ) }

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   <.cop 4183   class class class wbr 4653   {copab 4712   X. cxp 5112   ran crn 5115   |` cres 5116   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  dfres4  34061