Theorem nfmpt2 5593

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Hypotheses

Ref	Expression
nfmpt2.1	⊢ Ⅎ𝑧𝐴
nfmpt2.2	⊢ Ⅎ𝑧𝐵
nfmpt2.3	⊢ Ⅎ𝑧𝐶

Assertion

Ref	Expression
nfmpt2	⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

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

Proof of Theorem nfmpt2

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt2 5537	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		nfmpt2.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
3	2	nfcri 2213	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
4		nfmpt2.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
5	4	nfcri 2213	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
6	3, 5	nfan 1497	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7		nfmpt2.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
8	7	nfeq2 2230	. . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶
9	6, 8	nfan 1497	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)
10	9	nfoprab 5577	. 2 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
11	1, 10	nfcxfr 2216	1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 102   = wceq 1284   ∈ wcel 1433  Ⅎwnfc 2206  {coprab 5533   ↦ cmpt2 5534

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-oprab 5536  df-mpt2 5537

This theorem is referenced by:  nfiseq  9438