Theorem ssrel3 34067

Description: Subclass relation in another form when the subclass is a relation. (Contributed by Peter Mazsa, 16-Feb-2019.)

Assertion

Ref	Expression
ssrel3	|- ( Rel A -> ( A C_ B <-> A. x A. y ( x A y -> x B y ) ) )

Distinct variable groups:   x, A, y   x, B, y

Proof of Theorem ssrel3

Step	Hyp	Ref	Expression
1		ssrel 5207	. 2 |- ( Rel A -> ( A C_ B <-> A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) ) )
2		df-br 4654	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3		df-br 4654	. . . 4 |- ( x B y <-> <. x , y >. e. B )
4	2, 3	imbi12i 340	. . 3 |- ( ( x A y -> x B y ) <-> ( <. x , y >. e. A -> <. x , y >. e. B ) )
5	4	2albii 1748	. 2 |- ( A. x A. y ( x A y -> x B y ) <-> A. x A. y ( <. x , y >. e. A -> <. x , y >. e. B ) )
6	1, 5	syl6bbr 278	1 |- ( Rel A -> ( A C_ B <-> A. x A. y ( x A y -> x B y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   e. wcel 1990   C_ wss 3574   <.cop 4183   class class class wbr 4653   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  eqrel2  34068  inxpss  34082  inxpss2  34085