Theorem offveq 6256

Description: Convert an identity of the operation to the analogous identity on the function operation. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
offveq.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
offveq.2	⊢ (𝜑 → 𝐹 Fn 𝐴)
offveq.3	⊢ (𝜑 → 𝐺 Fn 𝐴)
offveq.4	⊢ (𝜑 → 𝐻 Fn 𝐴)
offveq.5	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
offveq.6	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
offveq.7	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))

Assertion

Ref	Expression
offveq	⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

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

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem offveq

Step	Hyp	Ref	Expression
1		offveq.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))
2	1	eqcomd 2237	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐻‘𝑥) = (𝐵𝑅𝐶))
3	2	ralrimiva 2605	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶))
4		offveq.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		offveq.2	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6		offveq.3	. . . 4 ⊢ (𝜑 → 𝐺 Fn 𝐴)
7		offveq.4	. . . 4 ⊢ (𝜑 → 𝐻 Fn 𝐴)
8		offveq.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
9		offveq.6	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
10	4, 5, 6, 7, 8, 9	offveqb 6255	. . 3 ⊢ (𝜑 → (𝐻 = (𝐹 ∘𝑓 𝑅𝐺) ↔ ∀𝑥 ∈ 𝐴 (𝐻‘𝑥) = (𝐵𝑅𝐶)))
11	3, 10	mpbird 167	. 2 ⊢ (𝜑 → 𝐻 = (𝐹 ∘𝑓 𝑅𝐺))
12	11	eqcomd 2237	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∀wral 2510   Fn wfn 5321  ‘cfv 5326  (class class class)co 6018   ∘_𝑓 cof 6233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-of 6235

This theorem is referenced by:  caofid0l  6262  caofid0r  6263  caofid1  6264  caofid2  6265