Theorem cnvss 4526

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 2993	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 3786	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 3786	. . . 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 4033	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4371	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4371	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3040	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 1433   C_ wss 2973   <.cop 3401   class class class wbr 3785   {copab 3838   `'ccnv 4362

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-br 3786  df-opab 3840  df-cnv 4371

This theorem is referenced by:  cnveq  4527  rnss  4582  relcnvtr  4860  funss  4940  funcnvuni  4988  funres11  4991  funcnvres  4992  foimacnv  5164  tposss  5884