Theorem ofcval 34256

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 2737	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
5	1, 2, 3, 4	ofcfval 34255	. . . 4 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
6	5	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
7		simpr 484	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7	fveq2d 6838	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	oveq1d 7373	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑋)𝑅𝐶))
10		simpr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		ovexd 7393	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) ∈ V)
12	6, 9, 10, 11	fvmptd 6948	. 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 7373	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
15	12, 14	eqtrd 2771	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ↦ cmpt 5179   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   ∘_f/c cofc 34252

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-ofc 34253

This theorem is referenced by:  probfinmeasb  34585