Theorem nfmpo1 7476

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 7401	. 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
2		nfoprab1 7457	. 2 ⊢ Ⅎ𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)}
3	1, 2	nfcxfr 2922	1 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ∧ wa 399   = wceq 1560   ∈ wcel 2142  Ⅎwnfc 2909  {coprab 7397   ∈ cmpo 7398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-oprab 7400  df-mpo 7401

This theorem is referenced by:  ovmpos  7544  ov2gf  7545  ovmpodxf  7546  ovmpodf  7552  ovmpodv2  7554  xpcomco  9039  mapxpen  9115  pwfseqlem2  10617  pwfseqlem4a  10619  pwfseqlem4  10620  gsum2d2lem  20013  gsum2d2  20014  gsumcom2  20015  dprd2d2  20086  cnmpt21  23731  cnmpt2t  23733  cnmptcom  23738  cnmpt2k  23748  xkocnv  23874  numclwlk2lem2f1o  30581  finxpreclem2  37884  mnringmulrcld  44804  fmuldfeqlem1  46158  fmuldfeq  46159  ovmpordxf  48961