Theorem nfmpo2 7514

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 7436	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab2 7495	. 2 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2903	1 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108  Ⅎwnfc 2890  {coprab 7432   ∈ cmpo 7433

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-oprab 7435  df-mpo 7436

This theorem is referenced by:  ovmpos  7581  ov2gf  7582  ovmpodxf  7583  ovmpodf  7589  ovmpodv2  7591  xpcomco  9102  mapxpen  9183  pwfseqlem2  10699  pwfseqlem4a  10701  pwfseqlem4  10702  gsum2d2lem  19991  gsum2d2  19992  gsumcom2  19993  dprd2d2  20064  cnmpt21  23679  cnmpt2t  23681  cnmptcom  23686  cnmpt2k  23696  xkocnv  23822  finxpreclem2  37391  finxpreclem6  37397  mnringmulrcld  44247  fmuldfeq  45598  smflimlem6  46791  ovmpordxf  48255