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Definition df-xrn 34476
Description: Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 34477 and brxrn 34479. This is Scott Fenton's df-txp 32299 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 34315 . 2 class (𝐴𝐵)
4 c1st 7314 . . . . . 6 class 1st
5 cvv 3351 . . . . . . 7 class V
65, 5cxp 5248 . . . . . 6 class (V × V)
74, 6cres 5252 . . . . 5 class (1st ↾ (V × V))
87ccnv 5249 . . . 4 class (1st ↾ (V × V))
98, 1ccom 5254 . . 3 class ((1st ↾ (V × V)) ∘ 𝐴)
10 c2nd 7315 . . . . . 6 class 2nd
1110, 6cres 5252 . . . . 5 class (2nd ↾ (V × V))
1211ccnv 5249 . . . 4 class (2nd ↾ (V × V))
1312, 2ccom 5254 . . 3 class ((2nd ↾ (V × V)) ∘ 𝐵)
149, 13cin 3723 . 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  34477  brxrn  34479  xrneq1  34482  xrneq2  34485  xrnres  34503  xrnres2  34504  xrnres3  34505
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