Definition df-xrn

Description: Define the range Cartesian product of two classes. Definition from Holmes p. 40. Membership in this class is characterized by xrnss3v and brxrn . This is Scott Fenton's df-txp with a different symbol, see https://github.com/metamath/set.mm/issues/2469 . (Contributed by Scott Fenton, 31-Mar-2012)

	Ref	Expression
Assertion	df-xrn	⊢ ( 𝐴 ⋉ 𝐵 ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐵 ) )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	⊢ 𝐴
1		cB	⊢ 𝐵
2	0 1	cxrn	⊢ ( 𝐴 ⋉ 𝐵 )
3		c1st	⊢ 1st
4		cvv	⊢ V
5	4 4	cxp	⊢ ( V × V )
6	3 5	cres	⊢ ( 1st ↾ ( V × V ) )
7	6	ccnv	⊢ ◡ ( 1st ↾ ( V × V ) )
8	7 0	ccom	⊢ ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐴 )
9		c2nd	⊢ 2nd
10	9 5	cres	⊢ ( 2nd ↾ ( V × V ) )
11	10	ccnv	⊢ ◡ ( 2nd ↾ ( V × V ) )
12	11 1	ccom	⊢ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐵 )
13	8 12	cin	⊢ ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐵 ) )
14	2 13	wceq	⊢ ( 𝐴 ⋉ 𝐵 ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝐵 ) )