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Theorem nfmpo 7244
Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpo.1 𝑧𝐴
nfmpo.2 𝑧𝐵
nfmpo.3 𝑧𝐶
Assertion
Ref Expression
nfmpo 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑦,𝑧)   𝐵(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem nfmpo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-mpo 7169 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
2 nfmpo.1 . . . . . 6 𝑧𝐴
32nfcri 2886 . . . . 5 𝑧 𝑥𝐴
4 nfmpo.2 . . . . . 6 𝑧𝐵
54nfcri 2886 . . . . 5 𝑧 𝑦𝐵
63, 5nfan 1905 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
7 nfmpo.3 . . . . 5 𝑧𝐶
87nfeq2 2916 . . . 4 𝑧 𝑤 = 𝐶
96, 8nfan 1905 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)
109nfoprab 7226 . 2 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
111, 10nfcxfr 2897 1 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1542  wcel 2113  wnfc 2879  {coprab 7165  cmpo 7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-oprab 7168  df-mpo 7169
This theorem is referenced by:  nfof  7424  el2mpocsbcl  7799  nfseq  13463  ptbasfi  22325  sdclem1  35513  fmuldfeqlem1  42649  stoweidlem51  43118  vonicc  43749
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