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Theorem nfmpt 4674
 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 4645 . 2 (𝑦𝐴𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐵)}
2 nfmpt.1 . . . . 5 𝑥𝐴
32nfcri 2745 . . . 4 𝑥 𝑦𝐴
4 nfmpt.2 . . . . 5 𝑥𝐵
54nfeq2 2766 . . . 4 𝑥 𝑧 = 𝐵
63, 5nfan 1816 . . 3 𝑥(𝑦𝐴𝑧 = 𝐵)
76nfopab 4650 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧 = 𝐵)}
81, 7nfcxfr 2749 1 𝑥(𝑦𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Ⅎwnfc 2738  {copab 4642   ↦ cmpt 4643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-opab 4644  df-mpt 4645 This theorem is referenced by:  ovmpt3rab1  6789  mpt2curryvald  7283  nfrdg  7397  mapxpen  8011  nfoi  8302  seqof2  12721  nfsum1  14268  nfsum  14269  fsumrlim  14384  fsumo1  14385  nfcprod1  14479  nfcprod  14480  gsum2d2  18196  prdsgsum  18200  dprd2d2  18266  gsumdixp  18432  mpfrcl  19339  ptbasfi  21194  ptcnplem  21234  ptcnp  21235  cnmptk2  21299  cnmpt2k  21301  xkocnv  21427  fsumcn  22481  itg2cnlem1  23334  nfitg  23347  itgfsum  23399  dvmptfsum  23542  itgulm2  23967  lgamgulm2  24562  fmptcof2  28839  fpwrelmap  28896  nfesum2  29430  sigapildsys  29552  oms0  29686  bnj1366  30154  poimirlem26  32605  cdleme32d  34750  cdleme32f  34752  cdlemksv2  35153  cdlemkuv2  35173  hlhilset  36244  aomclem8  36649  binomcxplemdvsum  37576  refsum2cn  38220  wessf1ornlem  38366  fmuldfeq  38650  fprodcnlem  38666  fprodcn  38667  fnlimfv  38730  fnlimcnv  38734  fnlimfvre  38741  fnlimfvre2  38744  fnlimf  38745  fnlimabslt  38746  fprodcncf  38787  dvnmptdivc  38828  dvmptfprod  38835  dvnprodlem1  38836  stoweidlem26  38919  stoweidlem31  38924  stoweidlem34  38927  stoweidlem35  38928  stoweidlem42  38935  stoweidlem48  38941  stoweidlem59  38952  fourierdlem31  39031  fourierdlem112  39111  sge0iunmptlemfi  39306  sge0iunmptlemre  39308  sge0iunmpt  39311  hoicvrrex  39446  ovncvrrp  39454  ovnhoilem1  39491  ovnlecvr2  39500  vonicc  39576  smflim  39663  smfmullem4  39679  aacllem  42356
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