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Theorem mptfvmpt 7248
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 6920 . . 3 𝑉 ∈ V
54mptex 7243 . 2 (𝑥𝑉𝐴) ∈ V
61, 2, 5fvmpt 7016 1 (𝑌𝑊 → (𝐺𝑌) = (𝑥𝑉𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  cmpt 5225  cfv 6561
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  cidfval  17719  idafval  18102  grpinvfvalALT  18997  grplactfval  19059  odfvalALT  19551  asclfval  21899  ig1pval  26215  ishlg  28610  htthlem  30936  sgnsv  33180  mvrsval  35510  mvhfval  35538  msrfval  35542  lkrfval  39088  pmapfval  39758  watfvalN  39994  ldilfset  40110  ltrnfset  40119  dilfsetN  40154  trnfsetN  40157  trlfset  40162  tgrpfset  40746  tendofset  40760  tendoi  40796  erngfset  40801  erngfset-rN  40809  dvafset  41006  diaffval  41032  dvhfset  41082  docaffvalN  41123  djaffvalN  41135  dibffval  41142  dicffval  41176  dihffval  41232  dihfval  41233  dochffval  41351  djhffval  41398  lcfrlem8  41551  lcdfval  41590  mapdffval  41628  mapdfval  41629  hvmapffval  41760  hdmap1ffval  41797  hdmapffval  41828  hdmapfval  41829  hgmapffval  41887  hgmapfval  41888  hbtlem1  43135  hbtlem7  43137
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