Theorem comffval2 17044

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffval2.o	⊢ 𝑂 = (compf‘𝐶)
comfffval2.b	⊢ 𝐵 = (Base‘𝐶)
comfffval2.h	⊢ 𝐻 = (Homf ‘𝐶)
comfffval2.x	⊢ · = (comp‘𝐶)
comffval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
comffval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
comffval2.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
comffval2	⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))

Distinct variable groups:   𝑓,𝑔,𝐵   𝐶,𝑓,𝑔   · ,𝑓,𝑔   𝑓,𝑋,𝑔   𝑓,𝑌,𝑔   𝜑,𝑓,𝑔   𝑓,𝑍,𝑔

Allowed substitution hints:   𝐻(𝑓,𝑔)   𝑂(𝑓,𝑔)

Proof of Theorem comffval2

Step	Hyp	Ref	Expression
1		comfffval2.o	. . 3 ⊢ 𝑂 = (compf‘𝐶)
2		comfffval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2758	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		comfffval2.x	. . 3 ⊢ · = (comp‘𝐶)
5		comffval2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		comffval2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		comffval2.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8	1, 2, 3, 4, 5, 6, 7	comffval 17041	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑍), 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
9		comfffval2.h	. . . 4 ⊢ 𝐻 = (Homf ‘𝐶)
10	9, 2, 3, 6, 7	homfval 17034	. . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
11	9, 2, 3, 5, 6	homfval 17034	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
12		eqidd 2759	. . 3 ⊢ (𝜑 → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓))
13	10, 11, 12	mpoeq123dv 7229	. 2 ⊢ (𝜑 → (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)) = (𝑔 ∈ (𝑌(Hom ‘𝐶)𝑍), 𝑓 ∈ (𝑋(Hom ‘𝐶)𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))
14	8, 13	eqtr4d 2796	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ⟨cop 4531  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  Basecbs 16555  Hom chom 16648  compcco 16649  Hom_f chomf 17009  comp_fccomf 17010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-homf 17013  df-comf 17014

This theorem is referenced by: (None)