Theorem fuco22nat 49241

Description: The composed natural transformation is a natural transformation. (Contributed by Zhi Wang, 2-Oct-2025.)

Hypotheses

Ref	Expression
fuco22nat.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco22nat.a	⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
fuco22nat.b	⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))
fuco22nat.u	⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)
fuco22nat.v	⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)

Assertion

Ref	Expression
fuco22nat	⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Proof of Theorem fuco22nat

Step	Hyp	Ref	Expression
1		fuco22nat.o	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
2		eqid 2730	. . 3 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22nat.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
4	2, 3	nat1st2nd 17922	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑀), (2nd ‘𝑀)⟩))
5		eqid 2730	. . 3 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22nat.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))
7	5, 6	nat1st2nd 17922	. 2 ⊢ (𝜑 → 𝐵 ∈ (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1st ‘𝑅), (2nd ‘𝑅)⟩))
8		fuco22nat.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)
9		relfunc 17830	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
10	5	natrcl 17921	. . . . . . 7 ⊢ (𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
11	6, 10	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
12	11	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
13		1st2nd 8027	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
14	9, 12, 13	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
15		relfunc 17830	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
16	2	natrcl 17921	. . . . . . 7 ⊢ (𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
17	3, 16	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
18	17	simpld 494	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
19		1st2nd 8027	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
20	15, 18, 19	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
21	14, 20	opeq12d 4853	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐹⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
22	8, 21	eqtrd 2765	. 2 ⊢ (𝜑 → 𝑈 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
23		fuco22nat.v	. . 3 ⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)
24	11	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (𝐷 Func 𝐸))
25		1st2nd 8027	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
26	9, 24, 25	sylancr 587	. . . 4 ⊢ (𝜑 → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
27	17	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func 𝐷))
28		1st2nd 8027	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
29	15, 27, 28	sylancr 587	. . . 4 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
30	26, 29	opeq12d 4853	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑀⟩ = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
31	23, 30	eqtrd 2765	. 2 ⊢ (𝜑 → 𝑉 = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
32	1, 4, 7, 22, 31	fuco22natlem 49240	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4603  Rel wrel 5651  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976   Func cfunc 17822   Nat cnat 17912   ∘_F cfuco 49211

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 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rmo 3357  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-riota 7351  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-map 8805  df-ixp 8875  df-cat 17635  df-cid 17636  df-func 17826  df-cofu 17828  df-nat 17914  df-fuco 49212

This theorem is referenced by:  fucof21  49242  fucocolem4  49251