Theorem ofcfval4 34399

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 6702	. . 3 ⊢ (𝜑 → dom 𝐹 = 𝐴)
3	2	mpteq1d 5190	. 2 ⊢ (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶)))
4		ofcfval4.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5	1, 4	fexd 7211	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
6		ofcfval4.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
7		ofcfval3 34396	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)))
8	5, 6, 7	syl2anc 593	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)))
9	1	ffvelcdmda 7065	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
10	1	feqmptd 6935	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
11		eqidd 2763	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)))
12		oveq1 7403	. . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥𝑅𝐶) = ((𝐹‘𝑦)𝑅𝐶))
13	9, 10, 11, 12	fmptco 7111	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶)))
14	3, 8, 13	3eqtr4d 2807	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  Vcvv 3454   ↦ cmpt 5181  dom cdm 5647   ∘ ccom 5651  ⟶wf 6517  ‘cfv 6521  (class class class)co 7396   ∘_f/c cofc 34389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-ofc 34390

This theorem is referenced by:  rrvmulc  34747