Theorem prelrrx2

Description: An unordered pair of ordered pairs with first components 1 and 2 and real numbers as second components is a point in a real Euclidean space of dimension 2. (Contributed by AV, 4-Feb-2023)

	Ref	Expression
Hypotheses	prelrrx2.i	|- I = { 1 , 2 }
	prelrrx2.b	|- P = ( RR ^m I )
Assertion	prelrrx2	|- ( ( A e. RR /\ B e. RR ) -> { <. 1 , A >. , <. 2 , B >. } e. P )

Proof

Step	Hyp	Ref	Expression
1		prelrrx2.i	|- I = { 1 , 2 }
2		prelrrx2.b	|- P = ( RR ^m I )
3		1ex	|- 1 e. _V
4		2ex	|- 2 e. _V
5	3 4	pm3.2i	|- ( 1 e. _V /\ 2 e. _V )
6	5	a1i	|- ( ( A e. RR /\ B e. RR ) -> ( 1 e. _V /\ 2 e. _V ) )
7		id	|- ( ( A e. RR /\ B e. RR ) -> ( A e. RR /\ B e. RR ) )
8		1ne2	|- 1 =/= 2
9	8	a1i	|- ( ( A e. RR /\ B e. RR ) -> 1 =/= 2 )
10	6 7 9	3jca	|- ( ( A e. RR /\ B e. RR ) -> ( ( 1 e. _V /\ 2 e. _V ) /\ ( A e. RR /\ B e. RR ) /\ 1 =/= 2 ) )
11		fprg	|- ( ( ( 1 e. _V /\ 2 e. _V ) /\ ( A e. RR /\ B e. RR ) /\ 1 =/= 2 ) -> { <. 1 , A >. , <. 2 , B >. } : { 1 , 2 } --> { A , B } )
12	10 11	syl	|- ( ( A e. RR /\ B e. RR ) -> { <. 1 , A >. , <. 2 , B >. } : { 1 , 2 } --> { A , B } )
13		prssi	|- ( ( A e. RR /\ B e. RR ) -> { A , B } C_ RR )
14	12 13	fssd	|- ( ( A e. RR /\ B e. RR ) -> { <. 1 , A >. , <. 2 , B >. } : { 1 , 2 } --> RR )
15		reex	|- RR e. _V
16		prex	|- { 1 , 2 } e. _V
17	15 16	pm3.2i	|- ( RR e. _V /\ { 1 , 2 } e. _V )
18		elmapg	|- ( ( RR e. _V /\ { 1 , 2 } e. _V ) -> ( { <. 1 , A >. , <. 2 , B >. } e. ( RR ^m { 1 , 2 } ) <-> { <. 1 , A >. , <. 2 , B >. } : { 1 , 2 } --> RR ) )
19	17 18	ax-mp	|- ( { <. 1 , A >. , <. 2 , B >. } e. ( RR ^m { 1 , 2 } ) <-> { <. 1 , A >. , <. 2 , B >. } : { 1 , 2 } --> RR )
20	14 19	sylibr	|- ( ( A e. RR /\ B e. RR ) -> { <. 1 , A >. , <. 2 , B >. } e. ( RR ^m { 1 , 2 } ) )
21	1	oveq2i	|- ( RR ^m I ) = ( RR ^m { 1 , 2 } )
22	2 21	eqtri	|- P = ( RR ^m { 1 , 2 } )
23	22	eleq2i	|- ( { <. 1 , A >. , <. 2 , B >. } e. P <-> { <. 1 , A >. , <. 2 , B >. } e. ( RR ^m { 1 , 2 } ) )
24	20 23	sylibr	|- ( ( A e. RR /\ B e. RR ) -> { <. 1 , A >. , <. 2 , B >. } e. P )