MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  frinxp Unicode version

Theorem frinxp 4771
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 3187 . . . . . . . . . . 11  |-  ( z 
C_  A  ->  (
x  e.  z  ->  x  e.  A )
)
2 ssel 3187 . . . . . . . . . . 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 4768 . . . . . . . . . . 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 2572 . . . . . 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 2573 . . . . 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 1556 . 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 4368 . 2  |-  ( R  Fr  A  <->  A. z
( ( z  C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x ) )
15 df-fr 4368 . 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 1530    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039    Fr wfr 4365    X. cxp 4703
This theorem is referenced by:  weinxp  4773
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-fr 4368  df-xp 4711
  Copyright terms: Public domain W3C validator