Theorem cnvin 5038

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.

Step	Hyp	Ref	Expression
1		df-cnv 4636	. . 3 |- `' ( A i^i B ) = { <. x , y >. | y ( A i^i B ) x }
2		inopab 4761	. . . 4 |- ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x /\ y B x ) }
3		brin 4057	. . . . 5 |- ( y ( A i^i B ) x <-> ( y A x /\ y B x ) )
4	3	opabbii 4072	. . . 4 |- { <. x , y >. | y ( A i^i B ) x } = { <. x , y >. | ( y A x /\ y B x ) }
5	2, 4	eqtr4i 2201	. . 3 |- ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } ) = { <. x , y >. | y ( A i^i B ) x }
6	1, 5	eqtr4i 2201	. 2 |- `' ( A i^i B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
7		df-cnv 4636	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4636	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	ineq12i 3336	. 2 |- ( `' A i^i `' B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2201	1 |- `' ( A i^i B ) = ( `' A i^i `' B )

Syntax hints:   /\ wa 104   = wceq 1353   i^i cin 3130   class class class wbr 4005   {copab 4065   `'ccnv 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-rel 4635  df-cnv 4636

This theorem is referenced by:  rnin  5040  dminxp  5075  imainrect  5076  cnvcnv  5083