Definition df-xrn 37546

Description: Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 37547 and brxrn 37549. This is Scott Fenton's df-txp 35128 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 35128. (Contributed by Scott Fenton, 31-Mar-2012.)

Assertion

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

Detailed syntax breakdown of Definition df-xrn

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cB	. . 3 class 𝐵
3	1, 2	cxrn 37347	. 2 class (𝐴 ⋉ 𝐵)
4		c1st 7977	. . . . . 6 class 1st
5		cvv 3472	. . . . . . 7 class V
6	5, 5	cxp 5675	. . . . . 6 class (V × V)
7	4, 6	cres 5679	. . . . 5 class (1st ↾ (V × V))
8	7	ccnv 5676	. . . 4 class ◡(1st ↾ (V × V))
9	8, 1	ccom 5681	. . 3 class (◡(1st ↾ (V × V)) ∘ 𝐴)
10		c2nd 7978	. . . . . 6 class 2nd
11	10, 6	cres 5679	. . . . 5 class (2nd ↾ (V × V))
12	11	ccnv 5676	. . . 4 class ◡(2nd ↾ (V × V))
13	12, 2	ccom 5681	. . 3 class (◡(2nd ↾ (V × V)) ∘ 𝐵)
14	9, 13	cin 3948	. 2 class ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
15	3, 14	wceq 1539	1 wff (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))

This definition is referenced by:  xrnss3v  37547  brxrn  37549  xrneq1  37552  xrneq2  37555  xrnres  37577  xrnres2  37578  xrnres3  37579