Theorem nfmpt22 5592

Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)

Assertion

Ref	Expression
nfmpt22	⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Proof of Theorem nfmpt22

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpt2 5537	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab2 5575	. 2 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2216	1 ⊢ Ⅎ𝑦(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-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:  ovmpt2s  5644  ov2gf  5645  ovmpt2dxf  5646  ovmpt2df  5652  ovmpt2dv2  5654  xpcomco  6323