Theorem ofcval 31358

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 2822	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
5	1, 2, 3, 4	ofcfval 31357	. . . 4 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
6	5	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
7		simpr 487	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7	fveq2d 6674	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	oveq1d 7171	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑋)𝑅𝐶))
10		simpr 487	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		ovexd 7191	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) ∈ V)
12	6, 9, 10, 11	fvmptd 6775	. 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 7171	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
15	12, 14	eqtrd 2856	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ↦ cmpt 5146   Fn wfn 6350  ‘cfv 6355  (class class class)co 7156   ∘_f/c cofc 31354

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-ofc 31355

This theorem is referenced by:  probfinmeasb  31686