Theorem qsss1 34053

Description: Subclass theorem for quotient sets. (Contributed by Peter Mazsa, 12-Sep-2020.)

Assertion

Ref	Expression
qsss1	|- ( A C_ B -> ( A /. C ) C_ ( B /. C ) )

Proof of Theorem qsss1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrexv 3667	. . 3 |- ( A C_ B -> ( E. x e. A y = [ x ] C -> E. x e. B y = [ x ] C ) )
2	1	ss2abdv 3675	. 2 |- ( A C_ B -> { y | E. x e. A y = [ x ] C } C_ { y | E. x e. B y = [ x ] C } )
3		df-qs 7748	. 2 |- ( A /. C ) = { y | E. x e. A y = [ x ] C }
4		df-qs 7748	. 2 |- ( B /. C ) = { y | E. x e. B y = [ x ] C }
5	2, 3, 4	3sstr4g 3646	1 |- ( A C_ B -> ( A /. C ) C_ ( B /. C ) )

Syntax hints:   -> wi 4   = wceq 1483   {cab 2608   E.wrex 2913   C_ wss 3574   [cec 7740   /.cqs 7741

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-nfc 2753  df-rex 2918  df-in 3581  df-ss 3588  df-qs 7748

This theorem is referenced by: (None)