Theorem mptfvmpt 7222

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 6887	. . 3 ⊢ 𝑉 ∈ V
5	4	mptex 7217	. 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V
6	1, 2, 5	fvmpt 6981	1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ↦ cmpt 5185  ‘cfv 6527

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  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 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  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 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535

This theorem is used by:  cidfval  17812  idafval  18194  grpinvfvalALT  19152  grplactfval  19213  odfvalALT  19709  asclfval  22148  ig1pval  26456  ishlg2  28999  ishlg  29002  htthlem  31453  sgnsv  33655  mvrsval  36191  mvhfval  36219  msrfval  36223  lkrfval  40064  pmapfval  40733  watfvalN  40969  ldilfset  41085  ltrnfset  41094  dilfsetN  41129  trnfsetN  41132  trlfset  41137  tgrpfset  41721  tendofset  41735  tendoi  41771  erngfset  41776  erngfset-rN  41784  dvafset  41981  diaffval  42007  dvhfset  42057  docaffvalN  42098  djaffvalN  42110  dibffval  42117  dicffval  42151  dihffval  42207  dihfval  42208  dochffval  42326  djhffval  42373  lcfrlem8  42526  lcdfval  42565  mapdffval  42603  mapdfval  42604  hvmapffval  42735  hdmap1ffval  42772  hdmapffval  42803  hdmapfval  42804  hgmapffval  42862  hgmapfval  42863  hbtlem1  44068  hbtlem7  44070