Definition df-mpo 5924

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 4093 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 5921	. 2 class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
7	1	cv 1363	. . . . . 6 class 𝑥
8	7, 3	wcel 2164	. . . . 5 wff 𝑥 ∈ 𝐴
9	2	cv 1363	. . . . . 6 class 𝑦
10	9, 4	wcel 2164	. . . . 5 wff 𝑦 ∈ 𝐵
11	8, 10	wa 104	. . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		vz	. . . . . 6 setvar 𝑧
13	12	cv 1363	. . . . 5 class 𝑧
14	13, 5	wceq 1364	. . . 4 wff 𝑧 = 𝐶
15	11, 14	wa 104	. . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)
16	15, 1, 2, 12	coprab 5920	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
17	6, 16	wceq 1364	1 wff (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}

This definition is referenced by:  mpoeq123  5978  mpoeq123dva  5980  mpoeq3dva  5983  nfmpo1  5986  nfmpo2  5987  nfmpo  5988  mpo0  5989  cbvmpox  5997  mpov  6009  mpomptx  6010  resmpo  6017  mpofun  6021  mpo2eqb  6029  rnmpo  6030  reldmmpo  6031  ovmpt4g  6042  elmpocl  6115  fmpox  6255  f1od2  6290  tposmpo  6336  erovlem  6683  xpcomco  6882  dfplpq2  7416  dfmpq2  7417  mpomulf  8011