Theorem ofcval 34835

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 34834	. . . 4 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
6	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹 ∘f/c 𝑅𝐶) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅𝐶)))
7		simpr 490	. . . . 5 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
8	7	fveq2d 6918	. . . 4 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → (𝐹‘𝑥) = (𝐹‘𝑋))
9	8	oveq1d 7464	. . 3 ⊢ (((𝜑 ∧ 𝑋 ∈ 𝐴) ∧ 𝑥 = 𝑋) → ((𝐹‘𝑥)𝑅𝐶) = ((𝐹‘𝑋)𝑅𝐶))
10		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ 𝐴)
11		ovexd 7484	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) ∈ V)
12	6, 9, 10, 11	fvmptd 7030	. 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 7464	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹‘𝑋)𝑅𝐶) = (𝐵𝑅𝐶))
15	12, 14	eqtrd 2798	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → ((𝐹 ∘f/c 𝑅𝐶)‘𝑋) = (𝐵𝑅𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5190   Fn wfn 6559  ‘cfv 6564  (class class class)co 7449   ∘_f/c cofc 34831

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-ov 7452  df-oprab 7453  df-mpo 7454  df-ofc 34832

This theorem is used by:  probfinmeasb  35164