Theorem nfmpt 5188

Description: Bound-variable hypothesis builder for the maps-to notation. (Contributed by NM, 20-Feb-2013.)

Hypotheses

Ref	Expression
nfmpt.1	⊢ Ⅎ𝑥𝐴
nfmpt.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfmpt	⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵)

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.

Step	Hyp	Ref	Expression
1		df-mpt 5165	. 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐵) = {⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
2		nfmpt.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2892	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfmpt.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfeq2 2922	. . . 4 ⊢ Ⅎ𝑥 𝑧 = 𝐵
6	3, 5	nfan 1900	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵)
7	6	nfopab 5150	. 2 ⊢ Ⅎ𝑥{⟨𝑦, 𝑧⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝑧 = 𝐵)}
8	1, 7	nfcxfr 2903	1 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ↦ 𝐵)

Syntax hints:   ∧ wa 397   = wceq 1539   ∈ wcel 2104  Ⅎwnfc 2885  {copab 5143   ↦ cmpt 5164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-opab 5144  df-mpt 5165

This theorem is referenced by:  ovmpt3rab1  7559  nfof  7571  mpocurryvald  8117  nfrdg  8276  mapxpen  8968  nfoi  9317  seqof2  13827  nfsum1  15446  nfsum  15447  nfsumOLD  15448  fsumrlim  15568  fsumo1  15569  nfcprod1  15665  nfcprod  15666  gsum2d2  19620  prdsgsum  19627  dprd2d2  19692  gsumdixp  19893  mpfrcl  21340  ptbasfi  22777  ptcnplem  22817  ptcnp  22818  cnmptk2  22882  cnmpt2k  22884  xkocnv  23010  fsumcn  24078  itg2cnlem1  24971  nfitg  24984  itgfsum  25036  dvmptfsum  25184  itgulm2  25613  lgamgulm2  26230  fmptcof2  31039  fpwrelmap  31113  nfesum2  32054  sigapildsys  32175  oms0  32309  bnj1366  32854  nosupbnd2  33964  noinfbnd2  33979  exrecfnlem  35594  poimirlem26  35847  cdleme32d  38500  cdleme32f  38502  cdlemksv2  38903  cdlemkuv2  38923  hlhilset  39990  pwsgprod  40306  aomclem8  40924  binomcxplemdvsum  42011  refsum2cn  42619  fmuldfeq  43173  fprodcnlem  43189  fprodcn  43190  fnlimfv  43253  fnlimcnv  43257  fnlimfvre  43264  fnlimfvre2  43267  fnlimf  43268  fnlimabslt  43269  fprodcncf  43490  dvnmptdivc  43528  dvmptfprod  43535  dvnprodlem1  43536  stoweidlem26  43616  stoweidlem31  43621  stoweidlem34  43624  stoweidlem35  43625  stoweidlem42  43632  stoweidlem48  43638  stoweidlem59  43649  fourierdlem31  43728  fourierdlem112  43808  sge0iunmptlemfi  44001  sge0iunmptlemre  44003  sge0iunmpt  44006  hoicvrrex  44144  ovncvrrp  44152  ovnhoilem1  44189  ovnlecvr2  44198  vonicc  44273  smflim  44365  smfmullem4  44382  smflim2  44393  smflimmpt  44397  smfsup  44401  smfsupmpt  44402  smfinf  44405  smfinfmpt  44406  smflimsuplem2  44408  smflimsuplem5  44411  smflimsup  44415  smflimsupmpt  44416  smfliminf  44418  smfliminfmpt  44419  aacllem  46563