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Theorem nfmpo1 7513
Description: Bound-variable hypothesis builder for an operation in maps-to notation. (Contributed by NM, 27-Aug-2013.)
Assertion
Ref Expression
nfmpo1 𝑥(𝑥𝐴, 𝑦𝐵𝐶)

Proof of Theorem nfmpo1
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-mpo 7436 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
2 nfoprab1 7494 . 2 𝑥{⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2nfcxfr 2901 1 𝑥(𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  wnfc 2888  {coprab 7432  cmpo 7433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-oprab 7435  df-mpo 7436
This theorem is referenced by:  ovmpos  7581  ov2gf  7582  ovmpodxf  7583  ovmpodf  7589  ovmpodv2  7591  xpcomco  9101  mapxpen  9182  pwfseqlem2  10697  pwfseqlem4a  10699  pwfseqlem4  10700  gsum2d2lem  20006  gsum2d2  20007  gsumcom2  20008  dprd2d2  20079  cnmpt21  23695  cnmpt2t  23697  cnmptcom  23702  cnmpt2k  23712  xkocnv  23838  numclwlk2lem2f1o  30408  finxpreclem2  37373  mnringmulrcld  44224  fmuldfeqlem1  45538  fmuldfeq  45539  ovmpordxf  48184
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