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Mirrors > Home > MPE Home > Th. List > Mathboxes > df-xrn | Structured version Visualization version GIF version |
Description: Define the range Cartesian product of two classes. Definition from [Holmes] p. 40. Membership in this class is characterized by xrnss3v 37237 and brxrn 37239. This is Scott Fenton's df-txp 34821 with a different symbol, see https://github.com/metamath/set.mm/issues/2469 34821. (Contributed by Scott Fenton, 31-Mar-2012.) |
Ref | Expression |
---|---|
df-xrn | ⊢ (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cB | . . 3 class 𝐵 | |
3 | 1, 2 | cxrn 37037 | . 2 class (𝐴 ⋉ 𝐵) |
4 | c1st 7972 | . . . . . 6 class 1st | |
5 | cvv 3474 | . . . . . . 7 class V | |
6 | 5, 5 | cxp 5674 | . . . . . 6 class (V × V) |
7 | 4, 6 | cres 5678 | . . . . 5 class (1st ↾ (V × V)) |
8 | 7 | ccnv 5675 | . . . 4 class ◡(1st ↾ (V × V)) |
9 | 8, 1 | ccom 5680 | . . 3 class (◡(1st ↾ (V × V)) ∘ 𝐴) |
10 | c2nd 7973 | . . . . . 6 class 2nd | |
11 | 10, 6 | cres 5678 | . . . . 5 class (2nd ↾ (V × V)) |
12 | 11 | ccnv 5675 | . . . 4 class ◡(2nd ↾ (V × V)) |
13 | 12, 2 | ccom 5680 | . . 3 class (◡(2nd ↾ (V × V)) ∘ 𝐵) |
14 | 9, 13 | cin 3947 | . 2 class ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) |
15 | 3, 14 | wceq 1541 | 1 wff (𝐴 ⋉ 𝐵) = ((◡(1st ↾ (V × V)) ∘ 𝐴) ∩ (◡(2nd ↾ (V × V)) ∘ 𝐵)) |
Colors of variables: wff setvar class |
This definition is referenced by: xrnss3v 37237 brxrn 37239 xrneq1 37242 xrneq2 37245 xrnres 37267 xrnres2 37268 xrnres3 37269 |
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