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Theorem cnvin 4826
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 4436 . . 3  |-  `' ( A  i^i  B )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
2 inopab 4556 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  /\  y B x ) }
3 brin 3884 . . . . 5  |-  ( y ( A  i^i  B
) x  <->  ( y A x  /\  y B x ) )
43opabbii 3897 . . . 4  |-  { <. x ,  y >.  |  y ( A  i^i  B
) x }  =  { <. x ,  y
>.  |  ( y A x  /\  y B x ) }
52, 4eqtr4i 2111 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
61, 5eqtr4i 2111 . 2  |-  `' ( A  i^i  B )  =  ( { <. x ,  y >.  |  y A x }  i^i  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4436 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4436 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8ineq12i 3197 . 2  |-  ( `' A  i^i  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2111 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    i^i cin 2996   class class class wbr 3837   {copab 3890   `'ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  rnin  4828  dminxp  4862  imainrect  4863  cnvcnv  4870
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