Theorem comfffn 17672

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 2730	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		eqid 2730	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17666	. 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 7423	. . 3 ⊢ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦) ∈ V
7		fvex 6874	. . 3 ⊢ ((Hom ‘𝐶)‘𝑥) ∈ V
8	6, 7	mpoex 8061	. 2 ⊢ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V
9	5, 8	fnmpoi 8052	1 ⊢ 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Syntax hints:   = wceq 1540   × cxp 5639   Fn wfn 6509  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  2^nd c2nd 7970  Basecbs 17186  Hom chom 17238  compcco 17239  comp_fccomf 17635

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-comf 17639

This theorem is referenced by: (None)