Theorem ofcfeqd2 33753

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 7433	. . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑅𝐶) = ((𝐹‘𝑥)𝑅𝐶))
2		oveq1 7433	. . . . 5 ⊢ (𝑦 = (𝐹‘𝑥) → (𝑦𝑃𝐶) = ((𝐹‘𝑥)𝑃𝐶))
3	1, 2	eqeq12d 2744	. . . 4 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝑦𝑅𝐶) = (𝑦𝑃𝐶) ↔ ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶)))
4		ofcfeqd2.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
5	4	ralrimiva 3143	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
6	5	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐵 (𝑦𝑅𝐶) = (𝑦𝑃𝐶))
7		ofcfeqd2.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)
8	3, 6, 7	rspcdva 3612	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑥)𝑃𝐶))
9	8	mpteq2dva 5252	. 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 2729	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
14	10, 11, 12, 13	ofcfval 33750	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
15	10, 11, 12, 13	ofcfval 33750	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑃𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑃𝐶)))
16	9, 14, 15	3eqtr4d 2778	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝐹 ∘f/c 𝑃𝐶))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058   ↦ cmpt 5235   Fn wfn 6548  ‘cfv 6553  (class class class)co 7426   ∘_f/c cofc 33747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-ofc 33748

This theorem is referenced by:  coinfliplem  34131