Theorem cnvss 4536

Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.)

Assertion

Ref	Expression
cnvss	|- ( A C_ B -> `' A C_ `' B )

Proof of Theorem cnvss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 2994	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 3794	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 3794	. . . 4 |- ( y B x <-> <. y , x >. e. B )
4	1, 2, 3	3imtr4g 203	. . 3 |- ( A C_ B -> ( y A x -> y B x ) )
5	4	ssopab2dv 4041	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4379	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4379	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3041	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 1434   C_ wss 2974   <.cop 3409   class class class wbr 3793   {copab 3846   `'ccnv 4370

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-in 2980  df-ss 2987  df-br 3794  df-opab 3848  df-cnv 4379

This theorem is referenced by:  cnveq  4537  rnss  4592  relcnvtr  4870  funss  4950  funcnvuni  4999  funres11  5002  funcnvres  5003  foimacnv  5175  tposss  5895