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Theorem cnvin 4718
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 4340 . . 3  |-  `' ( A  i^i  B )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
2 inopab 4455 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  /\  y B x ) }
3 brin 3808 . . . . 5  |-  ( y ( A  i^i  B
) x  <->  ( y A x  /\  y B x ) )
43opabbii 3821 . . . 4  |-  { <. x ,  y >.  |  y ( A  i^i  B
) x }  =  { <. x ,  y
>.  |  ( y A x  /\  y B x ) }
52, 4eqtr4i 2063 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
61, 5eqtr4i 2063 . 2  |-  `' ( A  i^i  B )  =  ( { <. x ,  y >.  |  y A x }  i^i  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4340 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4340 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8ineq12i 3133 . 2  |-  ( `' A  i^i  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2063 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    /\ wa 97    = wceq 1243    i^i cin 2913   class class class wbr 3761   {copab 3814   `'ccnv 4331
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3762  df-opab 3816  df-xp 4338  df-rel 4339  df-cnv 4340
This theorem is referenced by:  rnin  4720  dminxp  4752  imainrect  4753  cnvcnv  4760
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