Theorem fnfvof 7403

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 766	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐹 Fn 𝐴)
2		simplr 768	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐺 Fn 𝐴)
3		simpr 488	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
4		inidm 4145	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
5		eqidd 2799	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = (𝐹‘𝑋))
6		eqidd 2799	. . 3 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → (𝐺‘𝑋) = (𝐺‘𝑋))
7	1, 2, 3, 3, 4, 5, 6	ofval 7398	. 2 ⊢ ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ 𝐴 ∈ 𝑉) ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑋) = ((𝐹‘𝑋)𝑅(𝐺‘𝑋)))
8	7	anasss 470	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 399   = wceq 1538   ∈ wcel 2111   Fn wfn 6319  ‘cfv 6324  (class class class)co 7135   ∘_f cof 7387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389

This theorem is referenced by:  suppofssd  7850  ofccat  14320  ghmplusg  18959  lcomfsupp  19667  lmhmplusg  19809  frlmvplusgvalc  20456  frlmvscaval  20457  frlmsslsp  20485  frlmup1  20487  frlmup2  20488  islindf4  20527  evlslem3  20752  evlslem1  20754  coe1addfv  20894  evl1addd  20965  evl1subd  20966  evl1muld  20967  mamudi  21008  mamudir  21009  mdetrlin  21207  nmotri  23345  mdegaddle  24675  ply1rem  24764  fta1glem2  24767  fta1blem  24769  plyexmo  24909  ulmdvlem1  24995  jensen  25574  dchrmulcl  25833  dchrinv  25845  sumdchr2  25854  dchr2sum  25857  mzpsubst  39689  mzpcong  39913  rngunsnply  40117  lincsum  44838