Theorem nfmpo2 7473

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 7397	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab2 7454	. 2 ⊢ Ⅎ𝑦{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2921	1 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Ⅎwnfc 2908  {coprab 7393   ∈ cmpo 7394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ex 1799  df-nf 1803  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-oprab 7396  df-mpo 7397

This theorem is referenced by:  ovmpos  7540  ov2gf  7541  ovmpodxf  7542  ovmpodf  7548  ovmpodv2  7550  xpcomco  9035  mapxpen  9111  pwfseqlem2  10614  pwfseqlem4a  10616  pwfseqlem4  10617  gsum2d2lem  19996  gsum2d2  19997  gsumcom2  19998  dprd2d2  20069  cnmpt21  23711  cnmpt2t  23713  cnmptcom  23718  cnmpt2k  23728  xkocnv  23854  finxpreclem2  37848  finxpreclem6  37854  mnringmulrcld  44768  fmuldfeq  46123  smflimlem6  47314  ovmpordxf  48925