Theorem nfmpo2 7441

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 7365	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab2 7422	. 2 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2897	1 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  {coprab 7361   ∈ cmpo 7362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-oprab 7364  df-mpo 7365

This theorem is referenced by:  ovmpos  7508  ov2gf  7509  ovmpodxf  7510  ovmpodf  7516  ovmpodv2  7518  xpcomco  8998  mapxpen  9074  pwfseqlem2  10573  pwfseqlem4a  10575  pwfseqlem4  10576  gsum2d2lem  19939  gsum2d2  19940  gsumcom2  19941  dprd2d2  20012  cnmpt21  23646  cnmpt2t  23648  cnmptcom  23653  cnmpt2k  23663  xkocnv  23789  finxpreclem2  37720  finxpreclem6  37726  mnringmulrcld  44673  fmuldfeq  46031  smflimlem6  47222  ovmpordxf  48827