Theorem comffn 17628

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 2736	. . 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 7391	. . 3 ⊢ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓) ∈ V
3	1, 2	fnmpoi 8014	. 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 2736	. . . 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 17622	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)))
12	11	fneq1d 6585	. 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 2113  ⟨cop 4586   × cxp 5622   Fn wfn 6487  ‘cfv 6492  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17136  Hom chom 17188  compcco 17189  comp_fccomf 17590

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-comf 17594

This theorem is referenced by: (None)