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Theorem frinxp 4755
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 3174 . . . . . . . . . . 11  |-  ( z 
C_  A  ->  (
x  e.  z  ->  x  e.  A )
)
2 ssel 3174 . . . . . . . . . . 11  |-  ( z 
C_  A  ->  (
y  e.  z  -> 
y  e.  A ) )
31, 2anim12d 546 . . . . . . . . . 10  |-  ( z 
C_  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( x  e.  A  /\  y  e.  A ) ) )
4 brinxp 4752 . . . . . . . . . . 11  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
54ancoms 439 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
63, 5syl6 29 . . . . . . . . 9  |-  ( z 
C_  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( y R x  <->  y ( R  i^i  ( A  X.  A ) ) x ) ) )
76impl 603 . . . . . . . 8  |-  ( ( ( z  C_  A  /\  x  e.  z
)  /\  y  e.  z )  ->  (
y R x  <->  y ( R  i^i  ( A  X.  A ) ) x ) )
87notbid 285 . . . . . . 7  |-  ( ( ( z  C_  A  /\  x  e.  z
)  /\  y  e.  z )  ->  ( -.  y R x  <->  -.  y
( R  i^i  ( A  X.  A ) ) x ) )
98ralbidva 2559 . . . . . 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 2560 . . . . 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 451 . . . 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 236 . . 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 1553 . 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 4352 . 2  |-  ( R  Fr  A  <->  A. z
( ( z  C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x ) )
15 df-fr 4352 . 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 268 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 176    /\ wa 358   A.wal 1527    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    i^i cin 3151    C_ wss 3152   (/)c0 3455   class class class wbr 4023    Fr wfr 4349    X. cxp 4687
This theorem is referenced by:  weinxp  4757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-fr 4352  df-xp 4695
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