Theorem nfmpo1 7436

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

Assertion

Ref	Expression
nfmpo1	⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Proof of Theorem nfmpo1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 7361	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab1 7417	. 2 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2894	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2881  {coprab 7357   ∈ cmpo 7358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-oprab 7360  df-mpo 7361

This theorem is referenced by:  ovmpos  7504  ov2gf  7505  ovmpodxf  7506  ovmpodf  7512  ovmpodv2  7514  xpcomco  8993  mapxpen  9069  pwfseqlem2  10568  pwfseqlem4a  10570  pwfseqlem4  10571  gsum2d2lem  19900  gsum2d2  19901  gsumcom2  19902  dprd2d2  19973  cnmpt21  23613  cnmpt2t  23615  cnmptcom  23620  cnmpt2k  23630  xkocnv  23756  numclwlk2lem2f1o  30403  finxpreclem2  37534  mnringmulrcld  44411  fmuldfeqlem1  45770  fmuldfeq  45771  ovmpordxf  48527