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Theorem frinxp 4944
Description: Intersection of well-founded relation with cross 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.
StepHypRef Expression
1 ssel 3343 . . . . . . . . . . 11  |-  ( z 
C_  A  ->  (
x  e.  z  ->  x  e.  A )
)
2 ssel 3343 . . . . . . . . . . 11  |-  ( z 
C_  A  ->  (
y  e.  z  -> 
y  e.  A ) )
31, 2anim12d 548 . . . . . . . . . 10  |-  ( z 
C_  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( x  e.  A  /\  y  e.  A ) ) )
4 brinxp 4941 . . . . . . . . . . 11  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
54ancoms 441 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
63, 5syl6 32 . . . . . . . . 9  |-  ( z 
C_  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( y R x  <->  y ( R  i^i  ( A  X.  A ) ) x ) ) )
76impl 605 . . . . . . . 8  |-  ( ( ( z  C_  A  /\  x  e.  z
)  /\  y  e.  z )  ->  (
y R x  <->  y ( R  i^i  ( A  X.  A ) ) x ) )
87notbid 287 . . . . . . 7  |-  ( ( ( z  C_  A  /\  x  e.  z
)  /\  y  e.  z )  ->  ( -.  y R x  <->  -.  y
( R  i^i  ( A  X.  A ) ) x ) )
98ralbidva 2722 . . . . . 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 ) )
109rexbidva 2723 . . . . 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 ) )
1110adantr 453 . . . 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 ) )
1211pm5.74i 238 . . 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 ) )
1312albii 1576 . 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 4542 . 2  |-  ( R  Fr  A  <->  A. z
( ( z  C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x ) )
15 df-fr 4542 . 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 ) )
1613, 14, 153bitr4i 270 1  |-  ( R  Fr  A  <->  ( R  i^i  ( A  X.  A
) )  Fr  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550    e. wcel 1726    =/= wne 2600   A.wral 2706   E.wrex 2707    i^i cin 3320    C_ wss 3321   (/)c0 3629   class class class wbr 4213    Fr wfr 4539    X. cxp 4877
This theorem is referenced by:  weinxp  4946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-fr 4542  df-xp 4885
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