Theorem fuco22nat 49001

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 2734	. . 3 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22nat.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
4	2, 3	nat1st2nd 17971	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑀), (2nd ‘𝑀)⟩))
5		eqid 2734	. . 3 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22nat.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))
7	5, 6	nat1st2nd 17971	. 2 ⊢ (𝜑 → 𝐵 ∈ (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1st ‘𝑅), (2nd ‘𝑅)⟩))
8		fuco22nat.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)
9		relfunc 17879	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
10	5	natrcl 17970	. . . . . . 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 8047	. . . . 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 17879	. . . . 5 ⊢ Rel (𝐶 Func 𝐷)
16	2	natrcl 17970	. . . . . . 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 8047	. . . . 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 4863	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐹⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
22	8, 21	eqtrd 2769	. 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 8047	. . . . 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 8047	. . . . 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 4863	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑀⟩ = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
31	23, 30	eqtrd 2769	. 2 ⊢ (𝜑 → 𝑉 = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
32	1, 4, 7, 22, 31	fuco22natlem 49000	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) ∈ ((𝑂‘𝑈)(𝐶 Nat 𝐸)(𝑂‘𝑉)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ⟨cop 4614  Rel wrel 5672  ‘cfv 6542  (class class class)co 7414  1^st c1st 7995  2^nd c2nd 7996   Func cfunc 17871   Nat cnat 17961   ∘_F cfuco 48971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5261  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3773  df-csb 3882  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-iun 4975  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7997  df-2nd 7998  df-map 8851  df-ixp 8921  df-cat 17683  df-cid 17684  df-func 17875  df-cofu 17877  df-nat 17963  df-fuco 48972

This theorem is referenced by:  fucof21  49002  fucocolem4  49011