Theorem offveq 7071

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 3953	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
5	1, 2, 3, 3, 4	offn 7061	. 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 7059	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐵𝑅𝐶))
10		offveq.7	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐵𝑅𝐶) = (𝐻‘𝑥))
11	9, 10	eqtrd 2782	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹 ∘𝑓 𝑅𝐺)‘𝑥) = (𝐻‘𝑥))
12	5, 6, 11	eqfnfvd 6465	1 ⊢ (𝜑 → (𝐹 ∘𝑓 𝑅𝐺) = 𝐻)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1620   ∈ wcel 2127   Fn wfn 6032  ‘cfv 6037  (class class class)co 6801   ∘_𝑓 cof 7048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-of 7050

This theorem is referenced by:  caofid0l  7078  caofid0r  7079  caofid1  7080  caofid2  7081  ofnegsub  11181  bddibl  23776  dvaddf  23875  plydivlem3  24220  poimirlem5  33696  poimirlem10  33701  poimirlem22  33713  ofsubid  38994  ofmul12  38995  ofdivrec  38996  ofdivcan4  38997  ofdivdiv2  38998