Theorem comfffval2 17645

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 2733	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		comfffval2.x	. . 3 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17642	. 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 8008	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → (2nd ‘𝑥) ∈ 𝐵)
8	7	adantr 482	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑥) ∈ 𝐵)
9		simpr 486	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10	6, 2, 3, 8, 9	homfval 17636	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((2nd ‘𝑥)𝐻𝑦) = ((2nd ‘𝑥)(Hom ‘𝐶)𝑦))
11		xp1st 8007	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → (1st ‘𝑥) ∈ 𝐵)
12	11	adantr 482	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (1st ‘𝑥) ∈ 𝐵)
13	6, 2, 3, 12, 8	homfval 17636	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
14		df-ov 7412	. . . . . 6 ⊢ ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
15		df-ov 7412	. . . . . 6 ⊢ ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
16	13, 14, 15	3eqtr3g 2796	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
17		1st2nd2 8014	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
18	17	adantr 482	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
19	18	fveq2d 6896	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘𝑥) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
20	18	fveq2d 6896	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((Hom ‘𝐶)‘𝑥) = ((Hom ‘𝐶)‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
21	16, 19, 20	3eqtr4d 2783	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘𝑥) = ((Hom ‘𝐶)‘𝑥))
22		eqidd 2734	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
23	10, 21, 22	mpoeq123dv 7484	. . 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 7487	. 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 2764	1 ⊢ 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑥)𝐻𝑦), 𝑓 ∈ (𝐻‘𝑥) ↦ (𝑔(𝑥 · 𝑦)𝑓)))

Syntax hints:   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ⟨cop 4635   × cxp 5675  ‘cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1^st c1st 7973  2^nd c2nd 7974  Basecbs 17144  Hom chom 17208  compcco 17209  Hom_f chomf 17610  comp_fccomf 17611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-homf 17614  df-comf 17615

This theorem is referenced by: (None)