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Theorem nfmpt 4029
 Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)
Hypotheses
Ref Expression
nfmpt.1 𝑥𝐴
nfmpt.2 𝑥𝐵
Assertion
Ref Expression
nfmpt 𝑥(𝑦𝐴𝐵)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfmpt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-mpt 4000 . 2 (𝑦𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐵)}
2 nfmpt.1 . . . . 5 𝑥𝐴
32nfcri 2276 . . . 4 𝑥 𝑦𝐴
4 nfmpt.2 . . . . 5 𝑥𝐵
54nfeq2 2294 . . . 4 𝑥 𝑧 = 𝐵
63, 5nfan 1545 . . 3 𝑥(𝑦𝐴𝑧 = 𝐵)
76nfopab 4005 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐵)}
81, 7nfcxfr 2279 1 𝑥(𝑦𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332   ∈ wcel 1481  Ⅎwnfc 2269  {copab 3997   ↦ cmpt 3998 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-opab 3999  df-mpt 4000 This theorem is referenced by:  nfof  5996  nffrec  6302  mapxpen  6751  nfsum1  11177  nfsum  11178  nfcprod1  11375  nfcprod  11376  ctiunct  12009  fsumcncntop  12784  limcmpted  12860
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