Theorem fuco22nat 50398

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 2761	. . 3 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22nat.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
4	2, 3	nat1st2nd 18109	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑀), (2nd ‘𝑀)⟩))
5		eqid 2761	. . 3 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22nat.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))
7	5, 6	nat1st2nd 18109	. 2 ⊢ (𝜑 → 𝐵 ∈ (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1st ‘𝑅), (2nd ‘𝑅)⟩))
8		fuco22nat.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)
9		relfunc 18017	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
10	5	natrcl 18108	. . . . . . 7 ⊢ (𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
11	6, 10	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
12	11	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
13		1st2nd 8039	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
14	9, 12, 13	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
15		relfunc 18017	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
16	2	natrcl 18108	. . . . . . 7 ⊢ (𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
17	3, 16	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
18	17	simpld 500	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
19		1st2nd 8039	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
20	15, 18, 19	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
21	14, 20	opeq12d 4841	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐹⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
22	8, 21	eqtrd 2796	. 2 ⊢ (𝜑 → 𝑈 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
23		fuco22nat.v	. . 3 ⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)
24	11	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (𝐷 Func 𝐸))
25		1st2nd 8039	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
26	9, 24, 25	sylancr 599	. . . 4 ⊢ (𝜑 → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
27	17	simprd 501	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func 𝐷))
28		1st2nd 8039	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
29	15, 27, 28	sylancr 599	. . . 4 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
30	26, 29	opeq12d 4841	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑀⟩ = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
31	23, 30	eqtrd 2796	. 2 ⊢ (𝜑 → 𝑉 = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
32	1, 4, 7, 22, 31	fuco22natlem 50397	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⟨cop 4590  Rel wrel 5656  ‘cfv 6531  (class class class)co 7412  1^st c1st 7988  2^nd c2nd 7989   Func cfunc 18009   Nat cnat 18099   ∘_F cfuco 50368

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-map 8833  df-ixp 8910  df-cat 17822  df-cid 17823  df-func 18013  df-cofu 18015  df-nat 18101  df-fuco 50369

This theorem is used by:  fucof21  50399  fucocolem4  50408