Definition df-mpo 7372

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 𝐶(𝑥, 𝑦)". An extension of df-mpt 5168 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 7369	. 2 class (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)
7	1	cv 1541	. . . . . 6 class 𝑥
8	7, 3	wcel 2114	. . . . 5 wff 𝑥 ∈ 𝐴
9	2	cv 1541	. . . . . 6 class 𝑦
10	9, 4	wcel 2114	. . . . 5 wff 𝑦 ∈ 𝐵
11	8, 10	wa 395	. . . 4 wff (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
12		vz	. . . . . 6 setvar 𝑧
13	12	cv 1541	. . . . 5 class 𝑧
14	13, 5	wceq 1542	. . . 4 wff 𝑧 = 𝐶
15	11, 14	wa 395	. . 3 wff ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)
16	15, 1, 2, 12	coprab 7368	. 2 class {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
17	6, 16	wceq 1542	1 wff (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}

This definition is referenced by:  mpoeq123  7439  mpoeq123dva  7441  mpoeq3dva  7444  nfmpo1  7447  nfmpo2  7448  nfmpo  7449  0mpo0  7450  mpo0  7452  cbvmpox  7460  cbvmpov  7462  mpov  7479  mpomptx  7480  resmpo  7487  mpofun  7491  mpo2eqb  7499  rnmpo  7500  reldmmpo  7501  elrnmpores  7505  ovmpt4g  7514  mpondm0  7607  elmpocl  7608  fmpox  8020  bropopvvv  8040  bropfvvvv  8042  tposmpo  8213  erovlem  8760  xpcomco  9005  omxpenlem  9016  mpoaddf  11132  mpomulf  11133  cpnnen  16196  dmcuts  27783  mpomptxf  32751  df1stres  32777  df2ndres  32778  f1od2  32792  sxbrsigalem5  34432  cbvmpovw2  36424  cbvmpo1vw2  36425  cbvmpo2vw2  36426  cbvmpodavw2  36473  cbvmpo1davw2  36474  cbvmpo2davw2  36475  bj-dfmpoa  37430  csbmpo123  37647  uncf  37920  unccur  37924  mpobi123f  38483  cbvmpo2  45527  cbvmpo1  45528  mpomptx2  48805  cbvmpox2  48806  sectpropdlem  49505  invpropdlem  49507  isopropdlem  49509