Theorem mptfvmpt 7201

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 6870	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7196	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 6964	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136   ↦ cmpt 5175  ‘cfv 6510

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pr 5384

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518

This theorem is referenced by:  cidfval  17684  idafval  18066  grpinvfvalALT  18997  grplactfval  19059  odfvalALT  19549  asclfval  21903  ig1pval  26209  ishlg  28741  htthlem  31059  sgnsv  33294  mvrsval  35803  mvhfval  35831  msrfval  35835  lkrfval  39659  pmapfval  40328  watfvalN  40564  ldilfset  40680  ltrnfset  40689  dilfsetN  40724  trnfsetN  40727  trlfset  40732  tgrpfset  41316  tendofset  41330  tendoi  41366  erngfset  41371  erngfset-rN  41379  dvafset  41576  diaffval  41602  dvhfset  41652  docaffvalN  41693  djaffvalN  41705  dibffval  41712  dicffval  41746  dihffval  41802  dihfval  41803  dochffval  41921  djhffval  41968  lcfrlem8  42121  lcdfval  42160  mapdffval  42198  mapdfval  42199  hvmapffval  42330  hdmap1ffval  42367  hdmapffval  42398  hdmapfval  42399  hgmapffval  42457  hgmapfval  42458  hbtlem1  43648  hbtlem7  43650