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Theorem nfmpo 5847
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 5786 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
2 nfmpo.1 . . . . . 6 𝑧𝐴
32nfcri 2276 . . . . 5 𝑧 𝑥𝐴
4 nfmpo.2 . . . . . 6 𝑧𝐵
54nfcri 2276 . . . . 5 𝑧 𝑦𝐵
63, 5nfan 1545 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
7 nfmpo.3 . . . . 5 𝑧𝐶
87nfeq2 2294 . . . 4 𝑧 𝑤 = 𝐶
96, 8nfan 1545 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)
109nfoprab 5830 . 2 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
111, 10nfcxfr 2279 1 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1332  wcel 1481  wnfc 2269  {coprab 5782  cmpo 5783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-oprab 5785  df-mpo 5786
This theorem is referenced by:  nfof  5994  nfseq  10258
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