Definition df-xrn 39455

Description: Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 39456 and brxrn 39458. This is Scott Fenton's df-txp 36746 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 36746. (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 39249	. 2 class (𝐴 ⋉ 𝐵)
4		c1st 8009	. . . . . 6 class 1st
5		cvv 3454	. . . . . . 7 class V
6	5, 5	cxp 5653	. . . . . 6 class (V × V)
7	4, 6	cres 5657	. . . . 5 class (1st ↾ (V × V))
8	7	ccnv 5654	. . . 4 class ◡(1st ↾ (V × V))
9	8, 1	ccom 5659	. . 3 class (◡(1st ↾ (V × V)) ∘ 𝐴)
10		c2nd 8010	. . . . . 6 class 2nd
11	10, 6	cres 5657	. . . . 5 class (2nd ↾ (V × V))
12	11	ccnv 5654	. . . 4 class ◡(2nd ↾ (V × V))
13	12, 2	ccom 5659	. . 3 class (◡(2nd ↾ (V × V)) ∘ 𝐵)
14	9, 13	cin 4092	. 2 class ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))
15	3, 14	wceq 1569	1 wff (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵))

This definition is used by:  xrnss3v  39456  brxrn  39458  xrneq1  39471  xrneq2  39474  xrnres  39500  xrnres2  39501  xrnres3  39502