Definition df-mpo 5956

Description: Define maps-to notation for defining an operation via a rule. Read as "the operation defined by the map from 𝑥, 𝑦 (in 𝐴 × 𝐵) to 𝐵(𝑥, 𝑦)". An extension of df-mpt 4111 for two arguments. (Contributed by NM, 17-Feb-2008.)

Assertion

Ref	Expression
df-mpo	⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧   𝑧,𝐴   𝑧,𝐵   𝑧,𝐶

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)

Detailed syntax breakdown of Definition df-mpo

Step	Hyp	Ref	Expression
1		vx	. . 3 setvar 𝑥
2		vy	. . 3 setvar 𝑦
3		cA	. . 3 class 𝐴
4		cB	. . 3 class 𝐵
5		cC	. . 3 class 𝐶
6	1, 2, 3, 4, 5	cmpo 5953	. 2 class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
7	1	cv 1372	. . . . . 6 class 𝑥
8	7, 3	wcel 2177	. . . . 5 wff 𝑥 ∈ 𝐴
9	2	cv 1372	. . . . . 6 class 𝑦
10	9, 4	wcel 2177	. . . . 5 wff 𝑦 ∈ 𝐵
11	8, 10	wa 104	. . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		vz	. . . . . 6 setvar 𝑧
13	12	cv 1372	. . . . 5 class 𝑧
14	13, 5	wceq 1373	. . . 4 wff 𝑧 = 𝐶
15	11, 14	wa 104	. . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)
16	15, 1, 2, 12	coprab 5952	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
17	6, 16	wceq 1373	1 wff (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}

This definition is referenced by:  mpoeq123  6011  mpoeq123dva  6013  mpoeq3dva  6016  nfmpo1  6019  nfmpo2  6020  nfmpo  6021  mpo0  6022  cbvmpox  6030  mpov  6042  mpomptx  6043  resmpo  6050  mpofun  6054  mpo2eqb  6062  rnmpo  6063  reldmmpo  6064  ovmpt4g  6075  elmpocl  6148  fmpox  6293  f1od2  6328  tposmpo  6374  erovlem  6721  xpcomco  6928  dfplpq2  7474  dfmpq2  7475  mpomulf  8069