Theorem comfffval2 17665

Description: Value of the functionalized composition operation. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffval2.o	⊢ 𝑂 = (compf‘𝐶)
comfffval2.b	⊢ 𝐵 = (Base‘𝐶)
comfffval2.h	⊢ 𝐻 = (Homf ‘𝐶)
comfffval2.x	⊢ · = (comp‘𝐶)

Assertion

Ref	Expression
comfffval2	⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑦,𝐵   𝐶,𝑓,𝑔,𝑥,𝑦   · ,𝑓,𝑔,𝑥

Allowed substitution hints:   · (𝑦)   𝐻(𝑥,𝑦,𝑓,𝑔)   𝑂(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem comfffval2

Step	Hyp	Ref	Expression
1		comfffval2.o	. . 3 ⊢ 𝑂 = (compf‘𝐶)
2		comfffval2.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		eqid 2740	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		comfffval2.x	. . 3 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17662	. 2 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
6		comfffval2.h	. . . . 5 ⊢ 𝐻 = (Homf ‘𝐶)
7		xp2nd 7971	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → (2nd ‘𝑥) ∈ 𝐵)
8	7	adantr 481	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑥) ∈ 𝐵)
9		simpr 485	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10	6, 2, 3, 8, 9	homfval 17656	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((2nd ‘𝑥)𝐻𝑦) = ((2nd ‘𝑥)(Hom ‘𝐶)𝑦))
11		xp1st 7970	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → (1st ‘𝑥) ∈ 𝐵)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (1st ‘𝑥) ∈ 𝐵)
13	6, 2, 3, 12, 8	homfval 17656	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
14		df-ov 7366	. . . . . 6 ⊢ ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
15		df-ov 7366	. . . . . 6 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
16	13, 14, 15	3eqtr3g 2798	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
17		1st2nd2 7977	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
18	17	adantr 481	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
19	18	fveq2d 6838	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘𝑥) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
20	18	fveq2d 6838	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
21	16, 19, 20	3eqtr4d 2785	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘𝑥) = ((Hom ‘𝐶)‘𝑥))
22		eqidd 2741	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
23	10, 21, 22	mpoeq123dv 7438	. . 3 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)) = (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
24	23	mpoeq3ia 7441	. 2 ⊢ (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))
25	5, 24	eqtr4i 2766	1 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))

Syntax hints:   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ⟨cop 4568   × cxp 5623  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  1^st c1st 7936  2^nd c2nd 7937  Basecbs 17177  Hom chom 17229  compcco 17230  Hom_f chomf 17630  comp_fccomf 17631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-homf 17634  df-comf 17635

This theorem is referenced by: (None)