Theorem cnvin 4946

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 4547	. . 3 |- `' ( A i^i B ) = { <. x , y >. | y ( A i^i B ) x }
2		inopab 4671	. . . 4 |- ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x /\ y B x ) }
3		brin 3980	. . . . 5 |- ( y ( A i^i B ) x <-> ( y A x /\ y B x ) )
4	3	opabbii 3995	. . . 4 |- { <. x , y >. | y ( A i^i B ) x } = { <. x , y >. | ( y A x /\ y B x ) }
5	2, 4	eqtr4i 2163	. . 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 2163	. 2 |- `' ( A i^i B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
7		df-cnv 4547	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4547	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	ineq12i 3275	. 2 |- ( `' A i^i `' B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2163	1 |- `' ( A i^i B ) = ( `' A i^i `' B )

Syntax hints:   /\ wa 103   = wceq 1331   i^i cin 3070   class class class wbr 3929   {copab 3988   `'ccnv 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547

This theorem is referenced by:  rnin  4948  dminxp  4983  imainrect  4984  cnvcnv  4991