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Theorem cnvin 4916
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 4517 . . 3  |-  `' ( A  i^i  B )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
2 inopab 4641 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  /\  y B x ) }
3 brin 3950 . . . . 5  |-  ( y ( A  i^i  B
) x  <->  ( y A x  /\  y B x ) )
43opabbii 3965 . . . 4  |-  { <. x ,  y >.  |  y ( A  i^i  B
) x }  =  { <. x ,  y
>.  |  ( y A x  /\  y B x ) }
52, 4eqtr4i 2141 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
61, 5eqtr4i 2141 . 2  |-  `' ( A  i^i  B )  =  ( { <. x ,  y >.  |  y A x }  i^i  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4517 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4517 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8ineq12i 3245 . 2  |-  ( `' A  i^i  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2141 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1316    i^i cin 3040   class class class wbr 3899   {copab 3958   `'ccnv 4508
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by:  rnin  4918  dminxp  4953  imainrect  4954  cnvcnv  4961
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