Theorem comfffn 17610

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 2729	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2729	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17604	. 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 7382	. . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V
7		fvex 6835	. . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
8	6, 7	mpoex 8014	. 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V
9	5, 8	fnmpoi 8005	1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Syntax hints:   = wceq 1540   × cxp 5617   Fn wfn 6477  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  compcco 17173  comp_fccomf 17573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-comf 17577

This theorem is referenced by: (None)