Theorem comffn 17608

Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffn.o	⊢ 𝑂 = (compf‘𝐶)
comfffn.b	⊢ 𝐵 = (Base‘𝐶)
comffn.h	⊢ 𝐻 = (Hom ‘𝐶)
comffn.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
comffn.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
comffn.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
comffn	⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))

Proof of Theorem comffn

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2731	. . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓))
2		ovex 7379	. . 3 ⊢ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓) ∈ V
3	1, 2	fnmpoi 8002	. 2 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))
4		comfffn.o	. . . 4 ⊢ 𝑂 = (compf‘𝐶)
5		comfffn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
6		comffn.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
7		eqid 2731	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
8		comffn.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		comffn.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		comffn.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11	4, 5, 6, 7, 8, 9, 10	comffval 17602	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)))
12	11	fneq1d 6574	. 2 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) ↔ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))))
13	3, 12	mpbiri 258	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ⟨cop 4582   × cxp 5614   Fn wfn 6476  ‘cfv 6481  (class class class)co 7346   ∈ cmpo 7348  Basecbs 17117  Hom chom 17169  compcco 17170  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-comf 17574

This theorem is referenced by: (None)