Theorem frinxp 5184

Description: Intersection of well-founded relation with Cartesian product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
frinxp	|- ( R Fr A <-> ( R i^i ( A X. A ) ) Fr A )

Proof of Theorem frinxp

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3597	. . . . . . . . . . 11 |- ( z C_ A -> ( x e. z -> x e. A ) )
2		ssel 3597	. . . . . . . . . . 11 |- ( z C_ A -> ( y e. z -> y e. A ) )
3	1, 2	anim12d 586	. . . . . . . . . 10 |- ( z C_ A -> ( ( x e. z /\ y e. z ) -> ( x e. A /\ y e. A ) ) )
4		brinxp 5181	. . . . . . . . . . 11 |- ( ( y e. A /\ x e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
5	4	ancoms 469	. . . . . . . . . 10 |- ( ( x e. A /\ y e. A ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
6	3, 5	syl6 35	. . . . . . . . 9 |- ( z C_ A -> ( ( x e. z /\ y e. z ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) ) )
7	6	impl 650	. . . . . . . 8 |- ( ( ( z C_ A /\ x e. z ) /\ y e. z ) -> ( y R x <-> y ( R i^i ( A X. A ) ) x ) )
8	7	notbid 308	. . . . . . 7 |- ( ( ( z C_ A /\ x e. z ) /\ y e. z ) -> ( -. y R x <-> -. y ( R i^i ( A X. A ) ) x ) )
9	8	ralbidva 2985	. . . . . 6 |- ( ( z C_ A /\ x e. z ) -> ( A. y e. z -. y R x <-> A. y e. z -. y ( R i^i ( A X. A ) ) x ) )
10	9	rexbidva 3049	. . . . 5 |- ( z C_ A -> ( E. x e. z A. y e. z -. y R x <-> E. x e. z A. y e. z -. y ( R i^i ( A X. A ) ) x ) )
11	10	adantr 481	. . . 4 |- ( ( z C_ A /\ z =/= (/) ) -> ( E. x e. z A. y e. z -. y R x <-> E. x e. z A. y e. z -. y ( R i^i ( A X. A ) ) x ) )
12	11	pm5.74i 260	. . 3 |- ( ( ( z C_ A /\ z =/= (/) ) -> E. x e. z A. y e. z -. y R x ) <-> ( ( z C_ A /\ z =/= (/) ) -> E. x e. z A. y e. z -. y ( R i^i ( A X. A ) ) x ) )
13	12	albii 1747	. 2 |- ( A. z ( ( z C_ A /\ z =/= (/) ) -> E. x e. z A. y e. z -. y R x ) <-> A. z ( ( z C_ A /\ z =/= (/) ) -> E. x e. z A. y e. z -. y ( R i^i ( A X. A ) ) x ) )
14		df-fr 5073	. 2 |- ( R Fr A <-> A. z ( ( z C_ A /\ z =/= (/) ) -> E. x e. z A. y e. z -. y R x ) )
15		df-fr 5073	. 2 |- ( ( R i^i ( A X. A ) ) Fr A <-> A. z ( ( z C_ A /\ z =/= (/) ) -> E. x e. z A. y e. z -. y ( R i^i ( A X. A ) ) x ) )
16	13, 14, 15	3bitr4i 292	1 |- ( R Fr A <-> ( R i^i ( A X. A ) ) Fr A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Fr wfr 5070   X. cxp 5112

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-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-fr 5073  df-xp 5120

This theorem is referenced by:  weinxp  5186