Theorem comfffn 17761

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)

Assertion

Ref	Expression
comfffn	⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Proof of Theorem comfffn

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

Step	Hyp	Ref	Expression
1		comfffn.o	. . 3 ⊢ 𝑂 = (compf‘𝐶)
2		comfffn.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2763	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2763	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17755	. 2 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)))
6		ovex 7445	. . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V
7		fvex 6896	. . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
8	6, 7	mpoex 8077	. 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V
9	5, 8	fnmpoi 8068	1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Syntax hints:   = wceq 1570   × cxp 5661   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  2^nd c2nd 7986  Basecbs 17270  Hom chom 17322  compcco 17323  comp_fccomf 17724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-comf 17728

This theorem is referenced by: (None)