Theorem mptfvmpt 7226

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 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴))
mptfvmpt.g	⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀)
mptfvmpt.v	⊢ 𝑉 = (𝐹‘𝑋)

Assertion

Ref	Expression
mptfvmpt	⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

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

Step	Hyp	Ref	Expression
1		mptfvmpt.y	. 2 ⊢ (𝑦 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴))
2		mptfvmpt.g	. 2 ⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀)
3		mptfvmpt.v	. . . 4 ⊢ 𝑉 = (𝐹‘𝑋)
4	3	fvexi 6895	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7221	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 6989	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142   ↦ cmpt 5191  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544

This theorem is used by:  cidfval  17738  idafval  18120  grpinvfvalALT  19052  grplactfval  19113  odfvalALT  19609  asclfval  22039  ig1pval  26344  ishlg2  28882  ishlg  28885  htthlem  31280  sgnsv  33489  mvrsval  36005  mvhfval  36033  msrfval  36037  lkrfval  39889  pmapfval  40558  watfvalN  40794  ldilfset  40910  ltrnfset  40919  dilfsetN  40954  trnfsetN  40957  trlfset  40962  tgrpfset  41546  tendofset  41560  tendoi  41596  erngfset  41601  erngfset-rN  41609  dvafset  41806  diaffval  41832  dvhfset  41882  docaffvalN  41923  djaffvalN  41935  dibffval  41942  dicffval  41976  dihffval  42032  dihfval  42033  dochffval  42151  djhffval  42198  lcfrlem8  42351  lcdfval  42390  mapdffval  42428  mapdfval  42429  hvmapffval  42560  hdmap1ffval  42597  hdmapffval  42628  hdmapfval  42629  hgmapffval  42687  hgmapfval  42688  hbtlem1  43878  hbtlem7  43880