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Theorem frinxp 5074
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.
StepHypRef Expression
1 ssel 3493 . . . . . . . . . . 11  |-  ( z 
C_  A  ->  (
x  e.  z  ->  x  e.  A )
)
2 ssel 3493 . . . . . . . . . . 11  |-  ( z 
C_  A  ->  (
y  e.  z  -> 
y  e.  A ) )
31, 2anim12d 563 . . . . . . . . . 10  |-  ( z 
C_  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( x  e.  A  /\  y  e.  A ) ) )
4 brinxp 5071 . . . . . . . . . . 11  |-  ( ( y  e.  A  /\  x  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
54ancoms 453 . . . . . . . . . 10  |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( y R x  <-> 
y ( R  i^i  ( A  X.  A
) ) x ) )
63, 5syl6 33 . . . . . . . . 9  |-  ( z 
C_  A  ->  (
( x  e.  z  /\  y  e.  z )  ->  ( y R x  <->  y ( R  i^i  ( A  X.  A ) ) x ) ) )
76impl 620 . . . . . . . 8  |-  ( ( ( z  C_  A  /\  x  e.  z
)  /\  y  e.  z )  ->  (
y R x  <->  y ( R  i^i  ( A  X.  A ) ) x ) )
87notbid 294 . . . . . . 7  |-  ( ( ( z  C_  A  /\  x  e.  z
)  /\  y  e.  z )  ->  ( -.  y R x  <->  -.  y
( R  i^i  ( A  X.  A ) ) x ) )
98ralbidva 2893 . . . . . 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 2965 . . . . 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 465 . . . 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 245 . . 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 1641 . 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 4847 . 2  |-  ( R  Fr  A  <->  A. z
( ( z  C_  A  /\  z  =/=  (/) )  ->  E. x  e.  z  A. y  e.  z  -.  y R x ) )
15 df-fr 4847 . 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 277 1  |-  ( R  Fr  A  <->  ( R  i^i  ( A  X.  A
) )  Fr  A
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1393    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    i^i cin 3470    C_ wss 3471   (/)c0 3793   class class class wbr 4456    Fr wfr 4844    X. cxp 5006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-fr 4847  df-xp 5014
This theorem is referenced by:  weinxp  5076
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