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Theorem cnvin 4914
Description: Distributive law for converse over intersection. Theorem 15 of [Suppes] p. 62. (Contributed by NM, 25-Mar-1998.) (Revised by Mario Carneiro, 26-Jun-2014.)
Assertion
Ref Expression
cnvin  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )

Proof of Theorem cnvin
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 4515 . . 3  |-  `' ( A  i^i  B )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
2 inopab 4639 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  /\  y B x ) }
3 brin 3948 . . . . 5  |-  ( y ( A  i^i  B
) x  <->  ( y A x  /\  y B x ) )
43opabbii 3963 . . . 4  |-  { <. x ,  y >.  |  y ( A  i^i  B
) x }  =  { <. x ,  y
>.  |  ( y A x  /\  y B x ) }
52, 4eqtr4i 2139 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
61, 5eqtr4i 2139 . 2  |-  `' ( A  i^i  B )  =  ( { <. x ,  y >.  |  y A x }  i^i  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4515 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4515 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8ineq12i 3243 . 2  |-  ( `' A  i^i  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2139 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1314    i^i cin 3038   class class class wbr 3897   {copab 3956   `'ccnv 4506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515
This theorem is referenced by:  rnin  4916  dminxp  4951  imainrect  4952  cnvcnv  4959
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