Theorem comfffn 17748

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 2765	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2765	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17742	. 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 7433	. . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V
7		fvex 6884	. . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
8	6, 7	mpoex 8064	. 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V
9	5, 8	fnmpoi 8055	1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Syntax hints:   = wceq 1563   × cxp 5649   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  2^nd c2nd 7973  Basecbs 17257  Hom chom 17309  compcco 17310  comp_fccomf 17711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-comf 17715

This theorem is referenced by: (None)