Theorem br1steqg

Description: Uniqueness condition for the binary relation 1st . (Contributed by Scott Fenton, 2-Jul-2020) Revised to remove sethood hypothesis on C . (Revised by Peter Mazsa, 17-Jan-2022)

	Ref	Expression
Assertion	br1steqg	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. 1st C <-> C = A ) )

Proof

Step	Hyp	Ref	Expression
1		op1stg	|- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )
2	1	eqeq1d	|- ( ( A e. V /\ B e. W ) -> ( ( 1st ` <. A , B >. ) = C <-> A = C ) )
3		fo1st	|- 1st : _V -onto-> _V
4		fofn	|- ( 1st : _V -onto-> _V -> 1st Fn _V )
5	3 4	ax-mp	|- 1st Fn _V
6		opex	|- <. A , B >. e. _V
7		fnbrfvb	|- ( ( 1st Fn _V /\ <. A , B >. e. _V ) -> ( ( 1st ` <. A , B >. ) = C <-> <. A , B >. 1st C ) )
8	5 6 7	mp2an	|- ( ( 1st ` <. A , B >. ) = C <-> <. A , B >. 1st C )
9		eqcom	|- ( A = C <-> C = A )
10	2 8 9	3bitr3g	|- ( ( A e. V /\ B e. W ) -> ( <. A , B >. 1st C <-> C = A ) )