Theorem cnvss 4839

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 3177	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 4034	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 4034	. . . 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 4313	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4671	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4671	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3226	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 2167   C_ wss 3157   <.cop 3625   class class class wbr 4033   {copab 4093   `'ccnv 4662

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-br 4034  df-opab 4095  df-cnv 4671

This theorem is referenced by:  cnveq  4840  rnss  4896  relcnvtr  5189  funss  5277  funcnvuni  5327  funres11  5330  funcnvres  5331  foimacnv  5522  tposss  6304  structcnvcnv  12694  pw1nct  15647