Theorem nfmpo1 7492

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 7415	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab1 7473	. 2 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2897	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Ⅎwnfc 2884  {coprab 7411   ∈ cmpo 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-oprab 7414  df-mpo 7415

This theorem is referenced by:  ovmpos  7560  ov2gf  7561  ovmpodxf  7562  ovmpodf  7568  ovmpodv2  7570  xpcomco  9081  mapxpen  9162  pwfseqlem2  10678  pwfseqlem4a  10680  pwfseqlem4  10681  gsum2d2lem  19959  gsum2d2  19960  gsumcom2  19961  dprd2d2  20032  cnmpt21  23614  cnmpt2t  23616  cnmptcom  23621  cnmpt2k  23631  xkocnv  23757  numclwlk2lem2f1o  30365  finxpreclem2  37413  mnringmulrcld  44219  fmuldfeqlem1  45578  fmuldfeq  45579  ovmpordxf  48281