Theorem nqerrel

Description: Any member of ( N. X. N. ) relates to the representative of its equivalence class. (Contributed by Mario Carneiro, 6-May-2013) (New usage is discouraged.)

	Ref	Expression
Assertion	nqerrel	|- ( A e. ( N. X. N. ) -> A ~Q ( /Q ` A ) )

Proof

Step	Hyp	Ref	Expression
1		eqid	|- ( /Q ` A ) = ( /Q ` A )
2		nqerf	|- /Q : ( N. X. N. ) --> Q.
3		ffn	|- ( /Q : ( N. X. N. ) --> Q. -> /Q Fn ( N. X. N. ) )
4	2 3	ax-mp	|- /Q Fn ( N. X. N. )
5		fnbrfvb	|- ( ( /Q Fn ( N. X. N. ) /\ A e. ( N. X. N. ) ) -> ( ( /Q ` A ) = ( /Q ` A ) <-> A /Q ( /Q ` A ) ) )
6	4 5	mpan	|- ( A e. ( N. X. N. ) -> ( ( /Q ` A ) = ( /Q ` A ) <-> A /Q ( /Q ` A ) ) )
7	1 6	mpbii	|- ( A e. ( N. X. N. ) -> A /Q ( /Q ` A ) )
8		df-erq	|- /Q = ( ~Q i^i ( ( N. X. N. ) X. Q. ) )
9		inss1	|- ( ~Q i^i ( ( N. X. N. ) X. Q. ) ) C_ ~Q
10	8 9	eqsstri	|- /Q C_ ~Q
11	10	ssbri	|- ( A /Q ( /Q ` A ) -> A ~Q ( /Q ` A ) )
12	7 11	syl	|- ( A e. ( N. X. N. ) -> A ~Q ( /Q ` A ) )