Theorem fuco22nat 49821

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 2736	. . 3 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22nat.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
4	2, 3	nat1st2nd 17921	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑀), (2nd ‘𝑀)⟩))
5		eqid 2736	. . 3 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22nat.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))
7	5, 6	nat1st2nd 17921	. 2 ⊢ (𝜑 → 𝐵 ∈ (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1st ‘𝑅), (2nd ‘𝑅)⟩))
8		fuco22nat.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)
9		relfunc 17829	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
10	5	natrcl 17920	. . . . . . 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 7992	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
14	9, 12, 13	sylancr 588	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
15		relfunc 17829	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
16	2	natrcl 17920	. . . . . . 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 7992	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
20	15, 18, 19	sylancr 588	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
21	14, 20	opeq12d 4824	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐹⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
22	8, 21	eqtrd 2771	. 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 7992	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)) → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
26	9, 24, 25	sylancr 588	. . . 4 ⊢ (𝜑 → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
27	17	simprd 495	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func 𝐷))
28		1st2nd 7992	. . . . 5 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)) → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
29	15, 27, 28	sylancr 588	. . . 4 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
30	26, 29	opeq12d 4824	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑀⟩ = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
31	23, 30	eqtrd 2771	. 2 ⊢ (𝜑 → 𝑉 = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
32	1, 4, 7, 22, 31	fuco22natlem 49820	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⟨cop 4573  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941   Func cfunc 17821   Nat cnat 17911   ∘_F cfuco 49791

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-func 17825  df-cofu 17827  df-nat 17913  df-fuco 49792

This theorem is referenced by:  fucof21  49822  fucocolem4  49831