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Theorem cnvin 4755
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 4373 . . 3  |-  `' ( A  i^i  B )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
2 inopab 4490 . . . 4  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  ( y A x  /\  y B x ) }
3 brin 3834 . . . . 5  |-  ( y ( A  i^i  B
) x  <->  ( y A x  /\  y B x ) )
43opabbii 3847 . . . 4  |-  { <. x ,  y >.  |  y ( A  i^i  B
) x }  =  { <. x ,  y
>.  |  ( y A x  /\  y B x ) }
52, 4eqtr4i 2105 . . 3  |-  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )  =  { <. x ,  y >.  |  y ( A  i^i  B
) x }
61, 5eqtr4i 2105 . 2  |-  `' ( A  i^i  B )  =  ( { <. x ,  y >.  |  y A x }  i^i  {
<. x ,  y >.  |  y B x } )
7 df-cnv 4373 . . 3  |-  `' A  =  { <. x ,  y
>.  |  y A x }
8 df-cnv 4373 . . 3  |-  `' B  =  { <. x ,  y
>.  |  y B x }
97, 8ineq12i 3166 . 2  |-  ( `' A  i^i  `' B
)  =  ( {
<. x ,  y >.  |  y A x }  i^i  { <. x ,  y >.  |  y B x } )
106, 9eqtr4i 2105 1  |-  `' ( A  i^i  B )  =  ( `' A  i^i  `' B )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    i^i cin 2973   class class class wbr 3787   {copab 3840   `'ccnv 4364
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373
This theorem is referenced by:  rnin  4757  dminxp  4789  imainrect  4790  cnvcnv  4797
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