Theorem offveq 7636

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.2	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		offveq.3	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐴)
3		offveq.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
4		inidm 4174	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
5	1, 2, 3, 3, 4	offn 7623	. 2 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) Fn 𝐴)
6		offveq.4	. 2 ⊢ (𝜑 → 𝐻 Fn 𝐴)
7		offveq.5	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = 𝐵)
8		offveq.6	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐺‘𝑥) = 𝐶)
9	1, 2, 3, 3, 4, 7, 8	ofval 7621	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶))
10		offveq.7	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))
11	9, 10	eqtrd 2766	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘f 𝑅𝐺)‘𝑥) = (𝐻‘𝑥))
12	5, 6, 11	eqfnfvd 6967	1 ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   ∘_f cof 7608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-of 7610

This theorem is referenced by:  caofid0l  7643  caofid0r  7644  caofid1  7645  caofid2  7646  ofnegsub  12123  psdmul  22081  bddibl  25768  dvaddf  25872  plydivlem3  26230  poimirlem5  37675  poimirlem10  37680  poimirlem22  37692  fsuppssind  42696  ofsubid  44427  ofmul12  44428  ofdivrec  44429  ofdivcan4  44430  ofdivdiv2  44431