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Theorem nfmpo 5937
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 5873 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
2 nfmpo.1 . . . . . 6 𝑧𝐴
32nfcri 2313 . . . . 5 𝑧 𝑥𝐴
4 nfmpo.2 . . . . . 6 𝑧𝐵
54nfcri 2313 . . . . 5 𝑧 𝑦𝐵
63, 5nfan 1565 . . . 4 𝑧(𝑥𝐴𝑦𝐵)
7 nfmpo.3 . . . . 5 𝑧𝐶
87nfeq2 2331 . . . 4 𝑧 𝑤 = 𝐶
96, 8nfan 1565 . . 3 𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)
109nfoprab 5920 . 2 𝑧{⟨⟨𝑥, 𝑦⟩, 𝑤⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)}
111, 10nfcxfr 2316 1 𝑧(𝑥𝐴, 𝑦𝐵𝐶)
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  wnfc 2306  {coprab 5869  cmpo 5870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-oprab 5872  df-mpo 5873
This theorem is referenced by:  nfof  6081  nfseq  10428
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