Theorem cnvss 4835

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 3173	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 4030	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 4030	. . . 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 4309	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4667	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4667	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3222	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 2164   C_ wss 3153   <.cop 3621   class class class wbr 4029   {copab 4089   `'ccnv 4658

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3159  df-ss 3166  df-br 4030  df-opab 4091  df-cnv 4667

This theorem is referenced by:  cnveq  4836  rnss  4892  relcnvtr  5185  funss  5273  funcnvuni  5323  funres11  5326  funcnvres  5327  foimacnv  5518  tposss  6299  structcnvcnv  12634  pw1nct  15493