Theorem offvalfv 7645

Description: The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)

Hypotheses

Ref	Expression
offvalfv.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offvalfv.f	⊢ (𝜑 → 𝐹 Fn 𝐴)
offvalfv.g	⊢ (𝜑 → 𝐺 Fn 𝐴)

Assertion

Ref	Expression
offvalfv	⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝐺   𝜑,𝑥   𝑥,𝑅

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem offvalfv

Step	Hyp	Ref	Expression
1		offvalfv.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		offvalfv.f	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		fnfvelrn 7024	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
4	2, 3	sylan 587	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
5		offvalfv.g	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
6		fnfvelrn 7024	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
7	5, 6	sylan 587	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
8		dffn5 6888	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	2, 8	sylib 220	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
10		dffn5 6888	. . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
11	5, 10	sylib 220	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
12	1, 4, 7, 9, 11	offval2 7643	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121   ↦ cmpt 5155  ran crn 5621   Fn wfn 6483  ‘cfv 6488  (class class class)co 7359   ∘_f cof 7621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-of 7623

This theorem is referenced by:  mndpsuppss  18728  0mplrim  33708  mplvrpmmhm  33740  zlmodzxzscm  48860  zlmodzxzadd  48861  lincsum  48932