Theorem cnvss 4948

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 3242	. . . 4 |- ( A C_ B -> ( <. y , x >. e. A -> <. y , x >. e. B ) )
2		df-br 4126	. . . 4 |- ( y A x <-> <. y , x >. e. A )
3		df-br 4126	. . . 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 4416	. 2 |- ( A C_ B -> { <. x , y >. | y A x } C_ { <. x , y >. | y B x } )
6		df-cnv 4777	. 2 |- `' A = { <. x , y >. | y A x }
7		df-cnv 4777	. 2 |- `' B = { <. x , y >. | y B x }
8	5, 6, 7	3sstr4g 3291	1 |- ( A C_ B -> `' A C_ `' B )

Syntax hints:   -> wi 4   e. wcel 2209   C_ wss 3220   <.cop 3708   class class class wbr 4125   {copab 4186   `'ccnv 4768

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-br 4126  df-opab 4188  df-cnv 4777

This theorem is referenced by:  cnveq  4949  rnss  5007  relcnvtr  5302  funss  5391  funcnvuni  5445  funres11  5448  funcnvres  5449  foimacnv  5652  tposss  6507  structcnvcnv  13346  pw1nct  16947