Theorem offvalfv 7650

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 7030	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
4	2, 3	sylan 581	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ ran 𝐹)
5		offvalfv.g	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
6		fnfvelrn 7030	. . 3 ⊢ ((𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
7	5, 6	sylan 581	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) ∈ ran 𝐺)
8		dffn5 6896	. . 3 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
9	2, 8	sylib 218	. 2 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)))
10		dffn5 6896	. . 3 ⊢ (𝐺 Fn 𝐴 ↔ 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
11	5, 10	sylib 218	. 2 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐴 ↦ (𝐺‘𝑥)))
12	1, 4, 7, 9, 11	offval2 7648	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ↦ cmpt 5167  ran crn 5629   Fn wfn 6491  ‘cfv 6496  (class class class)co 7364   ∘_f cof 7626

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5523  df-xp 5634  df-rel 5635  df-cnv 5636  df-co 5637  df-dm 5638  df-rn 5639  df-res 5640  df-ima 5641  df-iota 6452  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7367  df-oprab 7368  df-mpo 7369  df-of 7628

This theorem is referenced by:  mndpsuppss  18730  mplvrpmmhm  33711  zlmodzxzscm  48853  zlmodzxzadd  48854  lincsum  48925