Theorem nfmpo2 7498

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

Assertion

Ref	Expression
nfmpo2	⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Proof of Theorem nfmpo2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mpo 7421	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab2 7479	. 2 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2890	1 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 394   = wceq 1534   ∈ wcel 2099  Ⅎwnfc 2876  {coprab 7417   ∈ cmpo 7418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-oprab 7420  df-mpo 7421

This theorem is referenced by:  ovmpos  7566  ov2gf  7567  ovmpodxf  7568  ovmpodf  7574  ovmpodv2  7576  xpcomco  9092  mapxpen  9173  pwfseqlem2  10693  pwfseqlem4a  10695  pwfseqlem4  10696  gsum2d2lem  19967  gsum2d2  19968  gsumcom2  19969  dprd2d2  20040  cnmpt21  23663  cnmpt2t  23665  cnmptcom  23670  cnmpt2k  23680  xkocnv  23806  finxpreclem2  37110  finxpreclem6  37116  mnringmulrcld  43939  fmuldfeq  45240  smflimlem6  46433  ovmpordxf  47753