Theorem cnvin 5011

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 4612	. . 3 |- `' ( A i^i B ) = { <. x , y >. | y ( A i^i B ) x }
2		inopab 4736	. . . 4 |- ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x /\ y B x ) }
3		brin 4034	. . . . 5 |- ( y ( A i^i B ) x <-> ( y A x /\ y B x ) )
4	3	opabbii 4049	. . . 4 |- { <. x , y >. | y ( A i^i B ) x } = { <. x , y >. | ( y A x /\ y B x ) }
5	2, 4	eqtr4i 2189	. . 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 2189	. 2 |- `' ( A i^i B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
7		df-cnv 4612	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4612	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	ineq12i 3321	. 2 |- ( `' A i^i `' B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2189	1 |- `' ( A i^i B ) = ( `' A i^i `' B )

Syntax hints:   /\ wa 103   = wceq 1343   i^i cin 3115   class class class wbr 3982   {copab 4042   `'ccnv 4603

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  rnin  5013  dminxp  5048  imainrect  5049  cnvcnv  5056