Theorem mptfvmpt 7202

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 6872	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7197	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 6968	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ↦ cmpt 5188  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  cidfval  17637  idafval  18019  grpinvfvalALT  18911  grplactfval  18973  odfvalALT  19463  asclfval  21788  ig1pval  26081  ishlg  28529  htthlem  30846  sgnsv  33117  mvrsval  35492  mvhfval  35520  msrfval  35524  lkrfval  39080  pmapfval  39750  watfvalN  39986  ldilfset  40102  ltrnfset  40111  dilfsetN  40146  trnfsetN  40149  trlfset  40154  tgrpfset  40738  tendofset  40752  tendoi  40788  erngfset  40793  erngfset-rN  40801  dvafset  40998  diaffval  41024  dvhfset  41074  docaffvalN  41115  djaffvalN  41127  dibffval  41134  dicffval  41168  dihffval  41224  dihfval  41225  dochffval  41343  djhffval  41390  lcfrlem8  41543  lcdfval  41582  mapdffval  41620  mapdfval  41621  hvmapffval  41752  hdmap1ffval  41789  hdmapffval  41820  hdmapfval  41821  hgmapffval  41879  hgmapfval  41880  hbtlem1  43112  hbtlem7  43114