Theorem cnvss 4903

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 3221	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 4089	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 4089	. . . 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 4373	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4733	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4733	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3270	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 2202   C_ wss 3200   <.cop 3672   class class class wbr 4088   {copab 4149   `'ccnv 4724

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-in 3206  df-ss 3213  df-br 4089  df-opab 4151  df-cnv 4733

This theorem is referenced by:  cnveq  4904  rnss  4962  relcnvtr  5256  funss  5345  funcnvuni  5399  funres11  5402  funcnvres  5403  foimacnv  5601  tposss  6411  structcnvcnv  13097  pw1nct  16604