Theorem nfmpt22 6621

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 6554	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab2 6603	. 2 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2749	1 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {coprab 6550   ↦ cmpt2 6551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-oprab 6553  df-mpt2 6554

This theorem is referenced by:  ovmpt2s  6682  ov2gf  6683  ovmpt2dxf  6684  ovmpt2df  6690  ovmpt2dv2  6692  xpcomco  7935  mapxpen  8011  pwfseqlem2  9360  pwfseqlem4a  9362  pwfseqlem4  9363  gsum2d2lem  18195  gsum2d2  18196  gsumcom2  18197  dprd2d2  18266  cnmpt21  21284  cnmpt2t  21286  cnmptcom  21291  cnmpt2k  21301  xkocnv  21427  finxpreclem2  32403  finxpreclem6  32409  fmuldfeq  38650  smflimlem6  39662  ovmpt2rdxf  41910