Theorem cnvss 4852

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 3187	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 4046	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 4046	. . . 4 |- ( y B x <-> <. y , x >. e. B )
4	1, 2, 3	3imtr4g 205	. . 3 |- ( A C_ B -> ( y A x -> y B x ) )
5	4	ssopab2dv 4326	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4684	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4684	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3236	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 2176   C_ wss 3166   <.cop 3636   class class class wbr 4045   {copab 4105   `'ccnv 4675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-br 4046  df-opab 4107  df-cnv 4684

This theorem is referenced by:  cnveq  4853  rnss  4909  relcnvtr  5203  funss  5291  funcnvuni  5344  funres11  5347  funcnvres  5348  foimacnv  5542  tposss  6334  structcnvcnv  12881  pw1nct  15977