Theorem nfmpt2 5604

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 5548	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
2		nfmpt2.1	. . . . . 6 ⊢ Ⅎ𝑧𝐴
3	2	nfcri 2214	. . . . 5 ⊢ Ⅎ𝑧 𝑥 ∈ 𝐴
4		nfmpt2.2	. . . . . 6 ⊢ Ⅎ𝑧𝐵
5	4	nfcri 2214	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ 𝐵
6	3, 5	nfan 1498	. . . 4 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
7		nfmpt2.3	. . . . 5 ⊢ Ⅎ𝑧𝐶
8	7	nfeq2 2231	. . . 4 ⊢ Ⅎ𝑧 𝑤 = 𝐶
9	6, 8	nfan 1498	. . 3 ⊢ Ⅎ𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)
10	9	nfoprab 5588	. 2 ⊢ Ⅎ𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑤 = 𝐶)}
11	1, 10	nfcxfr 2217	1 ⊢ Ⅎ𝑧(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 102   = wceq 1285   ∈ wcel 1434  Ⅎwnfc 2207  {coprab 5544   ↦ cmpt2 5545

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-oprab 5547  df-mpt2 5548

This theorem is referenced by:  nfiseq  9528