Theorem fuco22a 49005

Description: The morphism part of the functor composition bifunctor. See also fuco22 48994. (Contributed by Zhi Wang, 1-Oct-2025.)

Hypotheses

Ref	Expression
fuco22a.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco22a.u	⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)
fuco22a.v	⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)
fuco22a.a	⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
fuco22a.b	⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))

Assertion

Ref	Expression
fuco22a	⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘((1st ‘𝑀)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑀)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑅)‘((1st ‘𝑀)‘𝑥)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝐾)((1st ‘𝑀)‘𝑥))‘(𝐴‘𝑥)))))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸   𝑥,𝐹   𝑥,𝐾   𝑥,𝑀   𝑥,𝑅   𝑥,𝑈   𝑥,𝑉   𝜑,𝑥

Allowed substitution hints:   𝑃(𝑥)   𝑂(𝑥)

Proof of Theorem fuco22a

Step	Hyp	Ref	Expression
1		fuco22a.o	. 2 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
2		fuco22a.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨𝐾, 𝐹⟩)
3		relfunc 17879	. . . . . . 7 ⊢ Rel (𝐷 Func 𝐸)
4		df-rel 5674	. . . . . . 7 ⊢ (Rel (𝐷 Func 𝐸) ↔ (𝐷 Func 𝐸) ⊆ (V × V))
5	3, 4	mpbi 230	. . . . . 6 ⊢ (𝐷 Func 𝐸) ⊆ (V × V)
6		fuco22a.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))
7		eqid 2734	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
8	7	natrcl 17970	. . . . . . . 8 ⊢ (𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
9	6, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
10	9	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
11	5, 10	sselid 3963	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (V × V))
12		1st2ndb 8037	. . . . 5 ⊢ (𝐾 ∈ (V × V) ↔ 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
13	11, 12	sylib 218	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
14		relfunc 17879	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
15		df-rel 5674	. . . . . . 7 ⊢ (Rel (𝐶 Func 𝐷) ↔ (𝐶 Func 𝐷) ⊆ (V × V))
16	14, 15	mpbi 230	. . . . . 6 ⊢ (𝐶 Func 𝐷) ⊆ (V × V)
17		fuco22a.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
18		eqid 2734	. . . . . . . . 9 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
19	18	natrcl 17970	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
20	17, 19	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
21	20	simpld 494	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
22	16, 21	sselid 3963	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (V × V))
23		1st2ndb 8037	. . . . 5 ⊢ (𝐹 ∈ (V × V) ↔ 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
24	22, 23	sylib 218	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
25	13, 24	opeq12d 4863	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐹⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
26	2, 25	eqtrd 2769	. 2 ⊢ (𝜑 → 𝑈 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
27		fuco22a.v	. . 3 ⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)
28	9	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (𝐷 Func 𝐸))
29	5, 28	sselid 3963	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (V × V))
30		1st2ndb 8037	. . . . 5 ⊢ (𝑅 ∈ (V × V) ↔ 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
31	29, 30	sylib 218	. . . 4 ⊢ (𝜑 → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
32	20	simprd 495	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func 𝐷))
33	16, 32	sselid 3963	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (V × V))
34		1st2ndb 8037	. . . . 5 ⊢ (𝑀 ∈ (V × V) ↔ 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
35	33, 34	sylib 218	. . . 4 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
36	31, 35	opeq12d 4863	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑀⟩ = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
37	27, 36	eqtrd 2769	. 2 ⊢ (𝜑 → 𝑉 = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
38	18, 17	nat1st2nd 17971	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑀), (2nd ‘𝑀)⟩))
39	7, 6	nat1st2nd 17971	. 2 ⊢ (𝜑 → 𝐵 ∈ (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1st ‘𝑅), (2nd ‘𝑅)⟩))
40	1, 26, 37, 38, 39	fuco22 48994	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘((1st ‘𝑀)‘𝑥))(⟨((1st ‘𝐾)‘((1st ‘𝐹)‘𝑥)), ((1st ‘𝐾)‘((1st ‘𝑀)‘𝑥))⟩(comp‘𝐸)((1st ‘𝑅)‘((1st ‘𝑀)‘𝑥)))((((1st ‘𝐹)‘𝑥)(2nd ‘𝐾)((1st ‘𝑀)‘𝑥))‘(𝐴‘𝑥)))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  Vcvv 3464   ⊆ wss 3933  ⟨cop 4614   ↦ cmpt 5207   × cxp 5665  Rel wrel 5672  ‘cfv 6542  (class class class)co 7414  1^st c1st 7995  2^nd c2nd 7996  Basecbs 17230  compcco 17286   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-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-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7997  df-2nd 7998  df-ixp 8921  df-func 17875  df-cofu 17877  df-nat 17963  df-fuco 48972

This theorem is referenced by:  fucocolem2  49009  fucocolem4  49011  fucolid  49016  fucorid  49017  precofvalALT  49023