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Definition df-xrn 34475
Description: Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 34476 and brxrn 34478. This is Scott Fenton's df-txp 32298 with a different symbol, cf. https://github.com/metamath/set.mm/issues/2469. (Contributed by Scott Fenton, 31-Mar-2012.)
Assertion
Ref Expression
df-xrn (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))

Detailed syntax breakdown of Definition df-xrn
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2cxrn 34314 . 2 class (𝐴𝐵)
4 c1st 7313 . . . . . 6 class 1st
5 cvv 3351 . . . . . . 7 class V
65, 5cxp 5247 . . . . . 6 class (V × V)
74, 6cres 5251 . . . . 5 class (1st ↾ (V × V))
87ccnv 5248 . . . 4 class (1st ↾ (V × V))
98, 1ccom 5253 . . 3 class ((1st ↾ (V × V)) ∘ 𝐴)
10 c2nd 7314 . . . . . 6 class 2nd
1110, 6cres 5251 . . . . 5 class (2nd ↾ (V × V))
1211ccnv 5248 . . . 4 class (2nd ↾ (V × V))
1312, 2ccom 5253 . . 3 class ((2nd ↾ (V × V)) ∘ 𝐵)
149, 13cin 3722 . 2 class (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
153, 14wceq 1631 1 wff (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
Colors of variables: wff setvar class
This definition is referenced by:  xrnss3v  34476  brxrn  34478  xrneq1  34481  xrneq2  34484  xrnres  34502  xrnres2  34503  xrnres3  34504
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