Theorem fnfvof 7190

Description: Function value of a pointwise composition. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Proof shortened by Mario Carneiro, 5-Jun-2015.)

Assertion

Ref	Expression
fnfvof	⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))

Proof of Theorem fnfvof

Step	Hyp	Ref	Expression
1		simpll 757	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐹 Fn 𝐴)
2		simplr 759	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn 𝐴)
3		simpr 479	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
4		inidm 4043	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
5		eqidd 2779	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝐹‘𝑋))
6		eqidd 2779	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺‘𝑋))
7	1, 2, 3, 3, 4, 5, 6	ofval 7185	. 2 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
8	7	anasss 460	1 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝐴)) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107   Fn wfn 6132  ‘cfv 6137  (class class class)co 6924   ∘_𝑓 cof 7174

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pr 5140

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-of 7176

This theorem is referenced by:  ofccat  14121  ghmplusg  18639  lcomfsupp  19299  lmhmplusg  19443  evlslem3  19914  evlslem1  19915  coe1addfv  20035  evl1addd  20105  evl1subd  20106  evl1muld  20107  frlmvplusgvalc  20514  frlmvscaval  20515  frlmsslsp  20543  frlmup1  20545  frlmup2  20546  islindf4  20585  mamudi  20617  mamudir  20618  mdetrlin  20817  nmotri  22955  mdegaddle  24275  ply1rem  24364  fta1glem2  24367  fta1blem  24369  plyexmo  24509  ulmdvlem1  24595  jensen  25171  dchrmulcl  25430  dchrinv  25442  sumdchr2  25451  dchr2sum  25454  mzpsubst  38281  mzpcong  38508  rngunsnply  38712  lincsum  43243