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

Theorem mptfvmpt 7001
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 6688 . . 3 𝑉 ∈ V
54mptex 6996 . 2 (𝑥𝑉𝐴) ∈ V
61, 2, 5fvmpt 6775 1 (𝑌𝑊 → (𝐺𝑌) = (𝑥𝑉𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cmpt 5110  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347
This theorem is referenced by:  cidfval  17050  idafval  17429  grpinvfvalALT  18261  grplactfval  18318  odfvalALT  18779  asclfval  20692  ig1pval  24925  ishlg  26548  htthlem  28852  sgnsv  31004  mvrsval  33038  mvhfval  33066  msrfval  33070  lkrfval  36724  pmapfval  37393  watfvalN  37629  ldilfset  37745  ltrnfset  37754  dilfsetN  37789  trnfsetN  37792  trlfset  37797  tgrpfset  38381  tendofset  38395  tendoi  38431  erngfset  38436  erngfset-rN  38444  dvafset  38641  diaffval  38667  dvhfset  38717  docaffvalN  38758  djaffvalN  38770  dibffval  38777  dicffval  38811  dihffval  38867  dihfval  38868  dochffval  38986  djhffval  39033  lcfrlem8  39186  lcdfval  39225  mapdffval  39263  mapdfval  39264  hvmapffval  39395  hdmap1ffval  39432  hdmapffval  39463  hdmapfval  39464  hgmapffval  39522  hgmapfval  39523  hbtlem1  40520  hbtlem7  40522
  Copyright terms: Public domain W3C validator