Theorem cnvss 4802

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 3151	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 4006	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 4006	. . . 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 4280	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4636	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4636	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3200	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 2148   C_ wss 3131   <.cop 3597   class class class wbr 4005   {copab 4065   `'ccnv 4627

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-in 3137  df-ss 3144  df-br 4006  df-opab 4067  df-cnv 4636

This theorem is referenced by:  cnveq  4803  rnss  4859  relcnvtr  5150  funss  5237  funcnvuni  5287  funres11  5290  funcnvres  5291  foimacnv  5481  tposss  6249  structcnvcnv  12480  pw1nct  14837