Theorem ofcfval4 31368

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 6526	. . 3 ⊢ (𝜑 → dom 𝐹 = 𝐴)
3	2	mpteq1d 5158	. 2 ⊢ (𝜑 → (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶)))
4		ofcfval4.2	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		fex 6992	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
6	1, 4, 5	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐹 ∈ V)
7		ofcfval4.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑊)
8		ofcfval3 31365	. . 3 ⊢ ((𝐹 ∈ V ∧ 𝐶 ∈ 𝑊) → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)))
9	6, 7, 8	syl2anc 586	. 2 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = (𝑦 ∈ dom 𝐹 ↦ ((𝐹‘𝑦)𝑅𝐶)))
10	1	ffvelrnda 6854	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ 𝐵)
11	1	feqmptd 6736	. . 3 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ 𝐴 ↦ (𝐹‘𝑦)))
12		eqidd 2825	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) = (𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)))
13		oveq1 7166	. . 3 ⊢ (𝑥 = (𝐹‘𝑦) → (𝑥𝑅𝐶) = ((𝐹‘𝑦)𝑅𝐶))
14	10, 11, 12, 13	fmptco 6894	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹) = (𝑦 ∈ 𝐴 ↦ ((𝐹‘𝑦)𝑅𝐶)))
15	3, 9, 14	3eqtr4d 2869	1 ⊢ (𝜑 → (𝐹 ∘f/c 𝑅𝐶) = ((𝑥 ∈ 𝐵 ↦ (𝑥𝑅𝐶)) ∘ 𝐹))

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113  Vcvv 3497   ↦ cmpt 5149  dom cdm 5558   ∘ ccom 5562  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ∘_f/c cofc 31358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-ofc 31359

This theorem is referenced by:  rrvmulc  31715