Theorem mptfvmpt 7234

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 6899	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7229	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 6993	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ↦ cmpt 5185  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546

This theorem is used by:  cidfval  17851  idafval  18233  grpinvfvalALT  19191  grplactfval  19252  odfvalALT  19748  asclfval  22187  ig1pval  26495  ishlg2  29066  ishlg  29069  htthlem  31520  sgnsv  33764  mvrsval  36339  mvhfval  36367  msrfval  36371  lkrfval  40220  pmapfval  40889  watfvalN  41125  ldilfset  41241  ltrnfset  41250  dilfsetN  41285  trnfsetN  41288  trlfset  41293  tgrpfset  41877  tendofset  41891  tendoi  41927  erngfset  41932  erngfset-rN  41940  dvafset  42137  diaffval  42163  dvhfset  42213  docaffvalN  42254  djaffvalN  42266  dibffval  42273  dicffval  42307  dihffval  42363  dihfval  42364  dochffval  42482  djhffval  42529  lcfrlem8  42682  lcdfval  42721  mapdffval  42759  mapdfval  42760  hvmapffval  42891  hdmap1ffval  42928  hdmapffval  42959  hdmapfval  42960  hgmapffval  43018  hgmapfval  43019  hbtlem1  44201  hbtlem7  44203