Theorem mptfvmpt 7230

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 6896	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7225	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 6990	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5190  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is used by:  cidfval  17768  idafval  18150  grpinvfvalALT  19107  grplactfval  19168  odfvalALT  19664  asclfval  22097  ig1pval  26406  ishlg2  28945  ishlg  28948  htthlem  31399  sgnsv  33602  mvrsval  36086  mvhfval  36114  msrfval  36118  lkrfval  39962  pmapfval  40631  watfvalN  40867  ldilfset  40983  ltrnfset  40992  dilfsetN  41027  trnfsetN  41030  trlfset  41035  tgrpfset  41619  tendofset  41633  tendoi  41669  erngfset  41674  erngfset-rN  41682  dvafset  41879  diaffval  41905  dvhfset  41955  docaffvalN  41996  djaffvalN  42008  dibffval  42015  dicffval  42049  dihffval  42105  dihfval  42106  dochffval  42224  djhffval  42271  lcfrlem8  42424  lcdfval  42463  mapdffval  42501  mapdfval  42502  hvmapffval  42633  hdmap1ffval  42670  hdmapffval  42701  hdmapfval  42702  hgmapffval  42760  hgmapfval  42761  hbtlem1  43966  hbtlem7  43968