Theorem cnvin 4954

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 4555	. . 3 |- `' ( A i^i B ) = { <. x , y >. | y ( A i^i B ) x }
2		inopab 4679	. . . 4 |- ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x /\ y B x ) }
3		brin 3988	. . . . 5 |- ( y ( A i^i B ) x <-> ( y A x /\ y B x ) )
4	3	opabbii 4003	. . . 4 |- { <. x , y >. | y ( A i^i B ) x } = { <. x , y >. | ( y A x /\ y B x ) }
5	2, 4	eqtr4i 2164	. . 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 2164	. 2 |- `' ( A i^i B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
7		df-cnv 4555	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4555	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	ineq12i 3280	. 2 |- ( `' A i^i `' B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2164	1 |- `' ( A i^i B ) = ( `' A i^i `' B )

Syntax hints:   /\ wa 103   = wceq 1332   i^i cin 3075   class class class wbr 3937   {copab 3996   `'ccnv 4546

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555

This theorem is referenced by:  rnin  4956  dminxp  4991  imainrect  4992  cnvcnv  4999