Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfmpt21 Structured version   Visualization version   GIF version

Theorem nfmpt21 6687
 Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
Assertion
Ref Expression
nfmpt21 𝑥(𝑥𝐴, 𝑦𝐵𝐶)

Proof of Theorem nfmpt21
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-mpt2 6620 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 nfoprab1 6669 . 2 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2nfcxfr 2759 1 𝑥(𝑥𝐴, 𝑦𝐵𝐶)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Ⅎwnfc 2748  {coprab 6616   ↦ cmpt2 6617 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-oprab 6619  df-mpt2 6620 This theorem is referenced by:  ovmpt2s  6749  ov2gf  6750  ovmpt2dxf  6751  ovmpt2df  6757  ovmpt2dv2  6759  xpcomco  8010  mapxpen  8086  pwfseqlem2  9441  pwfseqlem4a  9443  pwfseqlem4  9444  gsum2d2lem  18312  gsum2d2  18313  gsumcom2  18314  dprd2d2  18383  cnmpt21  21414  cnmpt2t  21416  cnmptcom  21421  cnmpt2k  21431  xkocnv  21557  finxpreclem2  32898  fmuldfeqlem1  39250  fmuldfeq  39251  ovmpt2rdxf  41435
 Copyright terms: Public domain W3C validator