Theorem comffval2 17605

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 2731	. . 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 17602	. 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 17595	. . 3 ⊢ (𝜑 → (𝑌𝐻𝑍) = (𝑌(Hom ‘𝐶)𝑍))
11	9, 2, 3, 5, 6	homfval 17595	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝐶)𝑌))
12		eqidd 2732	. . 3 ⊢ (𝜑 → (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓))
13	10, 11, 12	mpoeq123dv 7421	. 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 2769	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩ · 𝑍)𝑓)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4582  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Basecbs 17117  Hom chom 17169  compcco 17170  Hom_f chomf 17569  comp_fccomf 17570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-homf 17573  df-comf 17574

This theorem is referenced by: (None)