Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptfvmpt Structured version   Visualization version   GIF version

Theorem mptfvmpt 6985
 Description: A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022.)
Hypotheses
Ref Expression
mptfvmpt.y (𝑦 = 𝑌𝑀 = (𝑥𝑉𝐴))
mptfvmpt.g 𝐺 = (𝑦𝑊𝑀)
mptfvmpt.v 𝑉 = (𝐹𝑋)
Assertion
Ref Expression
mptfvmpt (𝑌𝑊 → (𝐺𝑌) = (𝑥𝑉𝐴))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑉,𝑦   𝑦,𝑊   𝑦,𝑌
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑀(𝑥,𝑦)   𝑊(𝑥)   𝑋(𝑥,𝑦)   𝑌(𝑥)

Proof of Theorem mptfvmpt
StepHypRef Expression
1 mptfvmpt.y . 2 (𝑦 = 𝑌𝑀 = (𝑥𝑉𝐴))
2 mptfvmpt.g . 2 𝐺 = (𝑦𝑊𝑀)
3 mptfvmpt.v . . . 4 𝑉 = (𝐹𝑋)
43fvexi 6681 . . 3 𝑉 ∈ V
54mptex 6981 . 2 (𝑥𝑉𝐴) ∈ V
61, 2, 5fvmpt 6765 1 (𝑌𝑊 → (𝐺𝑌) = (𝑥𝑉𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1530   ∈ wcel 2107   ↦ cmpt 5143  ‘cfv 6352 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360 This theorem is referenced by:  cidfval  16937  idafval  17307  grpinvfvalALT  18073  grplactfval  18130  odfvalALT  18581  asclfval  20027  ig1pval  24684  ishlg  26305  htthlem  28611  sgnsv  30719  mvrsval  32639  mvhfval  32667  msrfval  32671  lkrfval  36093  pmapfval  36762  watfvalN  36998  ldilfset  37114  ltrnfset  37123  dilfsetN  37158  trnfsetN  37161  trlfset  37166  tgrpfset  37750  tendofset  37764  tendoi  37800  erngfset  37805  erngfset-rN  37813  dvafset  38010  diaffval  38036  dvhfset  38086  docaffvalN  38127  djaffvalN  38139  dibffval  38146  dicffval  38180  dihffval  38236  dihfval  38237  dochffval  38355  djhffval  38402  lcfrlem8  38555  lcdfval  38594  mapdffval  38632  mapdfval  38633  hvmapffval  38764  hdmap1ffval  38801  hdmapffval  38832  hdmapfval  38833  hgmapffval  38891  hgmapfval  38892  hbtlem1  39591  hbtlem7  39593
 Copyright terms: Public domain W3C validator