Theorem ofcval 34406

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 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) = (𝐹‘𝑥))
5	1, 2, 3, 4	ofcfval 34405	. . . 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 6875	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	oveq1d 7415	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑋)𝑅𝐶))
10		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		ovexd 7435	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) ∈ V)
12	6, 9, 10, 11	fvmptd 6987	. 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 7415	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
15	12, 14	eqtrd 2800	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ↦ cmpt 5186   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400   ∘_f/c cofc 34402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-ofc 34403

This theorem is referenced by:  probfinmeasb  34735