Theorem ofcfval4 31973

Description: The function/constant operation expressed as an operation composition. (Contributed by Thierry Arnoux, 31-Jan-2017.)

Hypotheses

Ref	Expression
ofcfval4.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
ofcfval4.2	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ofcfval4.3	⊢ (𝜑 → 𝐶 ∈ 𝑊)

Assertion

Ref	Expression
ofcfval4	⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹   𝑥,𝑅

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem ofcfval4

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofcfval4.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	fdmd 6595	. . 3 ⊢ (𝜑 → dom 𝐹 = 𝐴)
3	2	mpteq1d 5165	. 2 ⊢ (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶)))
4		ofcfval4.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	1, 4	fexd 7085	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
6		ofcfval4.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
7		ofcfval3 31970	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)))
8	5, 6, 7	syl2anc 583	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)))
9	1	ffvelrnda 6943	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
10	1	feqmptd 6819	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
11		eqidd 2739	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)))
12		oveq1 7262	. . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥𝑅𝐶) = ((𝐹‘𝑦)𝑅𝐶))
13	9, 10, 11, 12	fmptco 6983	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶)))
14	3, 8, 13	3eqtr4d 2788	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ↦ cmpt 5153  dom cdm 5580   ∘ ccom 5584  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ∘_f/c cofc 31963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-ofc 31964

This theorem is referenced by:  rrvmulc  32320