Theorem mptfvmpt 7265

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 6934	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7260	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 7029	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   ↦ cmpt 5249  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581

This theorem is referenced by:  cidfval  17734  idafval  18124  grpinvfvalALT  19019  grplactfval  19081  odfvalALT  19575  asclfval  21922  ig1pval  26235  ishlg  28628  htthlem  30949  sgnsv  33153  mvrsval  35473  mvhfval  35501  msrfval  35505  lkrfval  39043  pmapfval  39713  watfvalN  39949  ldilfset  40065  ltrnfset  40074  dilfsetN  40109  trnfsetN  40112  trlfset  40117  tgrpfset  40701  tendofset  40715  tendoi  40751  erngfset  40756  erngfset-rN  40764  dvafset  40961  diaffval  40987  dvhfset  41037  docaffvalN  41078  djaffvalN  41090  dibffval  41097  dicffval  41131  dihffval  41187  dihfval  41188  dochffval  41306  djhffval  41353  lcfrlem8  41506  lcdfval  41545  mapdffval  41583  mapdfval  41584  hvmapffval  41715  hdmap1ffval  41752  hdmapffval  41783  hdmapfval  41784  hgmapffval  41842  hgmapfval  41843  hbtlem1  43080  hbtlem7  43082