Theorem comfffn 16187

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 2610	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2610	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 16181	. 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 6577	. . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V
7		fvex 6113	. . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
8	6, 7	mpt2ex 7136	. 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V
9	5, 8	fnmpt2i 7128	1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Syntax hints:   = wceq 1475   × cxp 5036   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  2^nd c2nd 7058  Basecbs 15695  Hom chom 15779  compcco 15780  comp_fccomf 16151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-comf 16155

This theorem is referenced by: (None)