Theorem ofcfeqd2 34068

Description: Equality theorem for function/constant operation value. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcfeqd2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
ofcfeqd2.2	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
ofcfeqd2.3	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofcfeqd2.4	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcfeqd2.5	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
ofcfeqd2	⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f/c 𝑃𝐶))

Distinct variable groups:   𝑥,𝑦,𝐶   𝑥,𝐹,𝑦   𝑥,𝑃,𝑦   𝑥,𝑅,𝑦   𝜑,𝑥,𝑦   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem ofcfeqd2

Step	Hyp	Ref	Expression
1		oveq1 7356	. . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑅𝐶) = ((𝐹‘𝑥)𝑅𝐶))
2		oveq1 7356	. . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑃𝐶) = ((𝐹‘𝑥)𝑃𝐶))
3	1, 2	eqeq12d 2745	. . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶)))
4		ofcfeqd2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
5	4	ralrimiva 3121	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
7		ofcfeqd2.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
8	3, 6, 7	rspcdva 3578	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶))
9	8	mpteq2dva 5185	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶)))
10		ofcfeqd2.3	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
11		ofcfeqd2.4	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		ofcfeqd2.5	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
13		eqidd 2730	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
14	10, 11, 12, 13	ofcfval 34065	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
15	10, 11, 12, 13	ofcfval 34065	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑃𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶)))
16	9, 14, 15	3eqtr4d 2774	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f/c 𝑃𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ↦ cmpt 5173   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349   ∘_f/c cofc 34062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-ofc 34063

This theorem is referenced by:  coinfliplem  34447