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Theorem mptfvmpt 7179
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 6857 . . 3 𝑉 ∈ V
54mptex 7174 . 2 (𝑥𝑉𝐴) ∈ V
61, 2, 5fvmpt 6949 1 (𝑌𝑊 → (𝐺𝑌) = (𝑥𝑉𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cmpt 5189  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  cidfval  17561  idafval  17948  grpinvfvalALT  18795  grplactfval  18853  odfvalALT  19320  asclfval  21298  ig1pval  25553  ishlg  27586  htthlem  29901  sgnsv  32058  mvrsval  34156  mvhfval  34184  msrfval  34188  lkrfval  37595  pmapfval  38265  watfvalN  38501  ldilfset  38617  ltrnfset  38626  dilfsetN  38661  trnfsetN  38664  trlfset  38669  tgrpfset  39253  tendofset  39267  tendoi  39303  erngfset  39308  erngfset-rN  39316  dvafset  39513  diaffval  39539  dvhfset  39589  docaffvalN  39630  djaffvalN  39642  dibffval  39649  dicffval  39683  dihffval  39739  dihfval  39740  dochffval  39858  djhffval  39905  lcfrlem8  40058  lcdfval  40097  mapdffval  40135  mapdfval  40136  hvmapffval  40267  hdmap1ffval  40304  hdmapffval  40335  hdmapfval  40336  hgmapffval  40394  hgmapfval  40395  hbtlem1  41493  hbtlem7  41495
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