Theorem ofcval 34467

Description: Evaluate a function/constant operation at a point. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcfval.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
ofcfval.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcfval.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)
ofcval.6	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐵)

Assertion

Ref	Expression
ofcval	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

Proof of Theorem ofcval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcfval.1	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		ofcfval.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		ofcfval.3	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
4		eqidd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
5	1, 2, 3, 4	ofcfval 34466	. . . 4 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
6	5	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
7		simpr 489	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7	fveq2d 6887	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	oveq1d 7427	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑋)𝑅𝐶))
10		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		ovexd 7447	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) ∈ V)
12	6, 9, 10, 11	fvmptd 6999	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = ((𝐹‘𝑋)𝑅𝐶))
13		ofcval.6	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) = 𝐵)
14	13	oveq1d 7427	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
15	12, 14	eqtrd 2798	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ↦ cmpt 5193   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412   ∘_f/c cofc 34463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-ofc 34464

This theorem is referenced by:  probfinmeasb  34796