Theorem comfffval2 17644

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 2732	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		comfffval2.x	. . 3 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17641	. 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 8007	. . . . . 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 17635	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((2nd ‘𝑥)𝐻𝑦) = ((2nd ‘𝑥)(Hom ‘𝐶)𝑦))
11		xp1st 8006	. . . . . . . 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 17635	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
14		df-ov 7411	. . . . . 6 ⊢ ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
15		df-ov 7411	. . . . . 6 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
16	13, 14, 15	3eqtr3g 2795	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
17		1st2nd2 8013	. . . . . . 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 6895	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘𝑥) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
20	18	fveq2d 6895	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
21	16, 19, 20	3eqtr4d 2782	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘𝑥) = ((Hom ‘𝐶)‘𝑥))
22		eqidd 2733	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
23	10, 21, 22	mpoeq123dv 7483	. . 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 7486	. 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 2763	1 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))

Syntax hints:   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4634   × cxp 5674  ‘cfv 6543  (class class class)co 7408   ∈ cmpo 7410  1^st c1st 7972  2^nd c2nd 7973  Basecbs 17143  Hom chom 17207  compcco 17208  Hom_f chomf 17609  comp_fccomf 17610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7974  df-2nd 7975  df-homf 17613  df-comf 17614

This theorem is referenced by: (None)