Theorem mptfvmpt 7257

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 6922	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7252	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 7016	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147   ↦ cmpt 5189  ‘cfv 6558

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566

This theorem is used by:  cidfval  17879  idafval  18261  grpinvfvalALT  19219  grplactfval  19280  odfvalALT  19777  asclfval  22216  ig1pval  26524  ishlg2  29095  ishlg  29098  htthlem  31549  sgnsv  33791  mvrsval  36437  mvhfval  36465  msrfval  36469  lkrfval  40336  pmapfval  41005  watfvalN  41241  ldilfset  41357  ltrnfset  41366  dilfsetN  41401  trnfsetN  41404  trlfset  41409  tgrpfset  41993  tendofset  42007  tendoi  42043  erngfset  42048  erngfset-rN  42056  dvafset  42253  diaffval  42279  dvhfset  42329  docaffvalN  42370  djaffvalN  42382  dibffval  42389  dicffval  42423  dihffval  42479  dihfval  42480  dochffval  42598  djhffval  42645  lcfrlem8  42798  lcdfval  42837  mapdffval  42875  mapdfval  42876  hvmapffval  43007  hdmap1ffval  43044  hdmapffval  43075  hdmapfval  43076  hgmapffval  43134  hgmapfval  43135  hbtlem1  44317  hbtlem7  44319