Theorem comfffval2 17636

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 2737	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		comfffval2.x	. . 3 ⊢ · = (comp‘𝐶)
5	1, 2, 3, 4	comfffval 17633	. 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 7976	. . . . . 6 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → (2nd ‘𝑥) ∈ 𝐵)
8	7	adantr 480	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (2nd ‘𝑥) ∈ 𝐵)
9		simpr 484	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
10	6, 2, 3, 8, 9	homfval 17627	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((2nd ‘𝑥)𝐻𝑦) = ((2nd ‘𝑥)(Hom ‘𝐶)𝑦))
11		xp1st 7975	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → (1st ‘𝑥) ∈ 𝐵)
12	11	adantr 480	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (1st ‘𝑥) ∈ 𝐵)
13	6, 2, 3, 12, 8	homfval 17627	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = ((1st ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑥)))
14		df-ov 7371	. . . . . 6 ⊢ ((1st ‘𝑥)𝐻(2nd ‘𝑥)) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
15		df-ov 7371	. . . . . 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 7982	. . . . . . 7 ⊢ (𝑥 ∈ (𝐵 × 𝐵) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
18	17	adantr 480	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → 𝑥 = ⟨(1st ‘𝑥), (2nd ‘𝑥)⟩)
19	18	fveq2d 6846	. . . . 5 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝐻‘𝑥) = (𝐻‘⟨(1st ‘𝑥), (2nd ‘𝑥)⟩))
20	18	fveq2d 6846	. . . . 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 2738	. . . 4 ⊢ ((𝑥 ∈ (𝐵 × 𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑔(𝑥 · 𝑦)𝑓) = (𝑔(𝑥 · 𝑦)𝑓))
23	10, 21, 22	mpoeq123dv 7443	. . 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 7446	. 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 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4588   × cxp 5630  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1^st c1st 7941  2^nd c2nd 7942  Basecbs 17148  Hom chom 17200  compcco 17201  Hom_f chomf 17601  comp_fccomf 17602

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-homf 17605  df-comf 17606

This theorem is referenced by: (None)