Theorem cnvin 4731

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 4353	. . 3 |- `' ( A i^i B ) = { <. x , y >. | y ( A i^i B ) x }
2		inopab 4468	. . . 4 |- ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } ) = { <. x , y >. | ( y A x /\ y B x ) }
3		brin 3811	. . . . 5 |- ( y ( A i^i B ) x <-> ( y A x /\ y B x ) )
4	3	opabbii 3824	. . . 4 |- { <. x , y >. | y ( A i^i B ) x } = { <. x , y >. | ( y A x /\ y B x ) }
5	2, 4	eqtr4i 2063	. . 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 2063	. 2 |- `' ( A i^i B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
7		df-cnv 4353	. . 3 |- `' A = { <. x , y >. | y A x }
8		df-cnv 4353	. . 3 |- `' B = { <. x , y >. | y B x }
9	7, 8	ineq12i 3136	. 2 |- ( `' A i^i `' B ) = ( { <. x , y >. | y A x } i^i { <. x , y >. | y B x } )
10	6, 9	eqtr4i 2063	1 |- `' ( A i^i B ) = ( `' A i^i `' B )

Syntax hints:   /\ wa 97   = wceq 1243   i^i cin 2916   class class class wbr 3764   {copab 3817   `'ccnv 4344

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353

This theorem is referenced by:  rnin  4733  dminxp  4765  imainrect  4766  cnvcnv  4773