Theorem fuco22a 50047

Description: The morphism part of the functor composition bifunctor. See also fuco22 50036. (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 17919	. . . . . . 7 ⊢ Rel (𝐷 Func 𝐸)
4		df-rel 5669	. . . . . . 7 ⊢ (Rel (𝐷 Func 𝐸) ↔ (𝐷 Func 𝐸) ⊆ (V × V))
5	3, 4	mpbi 233	. . . . . 6 ⊢ (𝐷 Func 𝐸) ⊆ (V × V)
6		fuco22a.b	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅))
7		eqid 2769	. . . . . . . . 9 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
8	7	natrcl 18010	. . . . . . . 8 ⊢ (𝐵 ∈ (𝐾(𝐷 Nat 𝐸)𝑅) → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
9	6, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐾 ∈ (𝐷 Func 𝐸) ∧ 𝑅 ∈ (𝐷 Func 𝐸)))
10	9	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
11	5, 10	sselid 3943	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (V × V))
12		1st2ndb 8026	. . . . 5 ⊢ (𝐾 ∈ (V × V) ↔ 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
13	11, 12	sylib 221	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
14		relfunc 17919	. . . . . . 7 ⊢ Rel (𝐶 Func 𝐷)
15		df-rel 5669	. . . . . . 7 ⊢ (Rel (𝐶 Func 𝐷) ↔ (𝐶 Func 𝐷) ⊆ (V × V))
16	14, 15	mpbi 233	. . . . . 6 ⊢ (𝐶 Func 𝐷) ⊆ (V × V)
17		fuco22a.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀))
18		eqid 2769	. . . . . . . . 9 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
19	18	natrcl 18010	. . . . . . . 8 ⊢ (𝐴 ∈ (𝐹(𝐶 Nat 𝐷)𝑀) → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
20	17, 19	syl 18	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ (𝐶 Func 𝐷) ∧ 𝑀 ∈ (𝐶 Func 𝐷)))
21	20	simpld 499	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
22	16, 21	sselid 3943	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (V × V))
23		1st2ndb 8026	. . . . 5 ⊢ (𝐹 ∈ (V × V) ↔ 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
24	22, 23	sylib 221	. . . 4 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
25	13, 24	opeq12d 4850	. . 3 ⊢ (𝜑 → ⟨𝐾, 𝐹⟩ = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
26	2, 25	eqtrd 2804	. 2 ⊢ (𝜑 → 𝑈 = ⟨⟨(1st ‘𝐾), (2nd ‘𝐾)⟩, ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩⟩)
27		fuco22a.v	. . 3 ⊢ (𝜑 → 𝑉 = ⟨𝑅, 𝑀⟩)
28	9	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (𝐷 Func 𝐸))
29	5, 28	sselid 3943	. . . . 5 ⊢ (𝜑 → 𝑅 ∈ (V × V))
30		1st2ndb 8026	. . . . 5 ⊢ (𝑅 ∈ (V × V) ↔ 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
31	29, 30	sylib 221	. . . 4 ⊢ (𝜑 → 𝑅 = ⟨(1st ‘𝑅), (2nd ‘𝑅)⟩)
32	20	simprd 500	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐶 Func 𝐷))
33	16, 32	sselid 3943	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (V × V))
34		1st2ndb 8026	. . . . 5 ⊢ (𝑀 ∈ (V × V) ↔ 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
35	33, 34	sylib 221	. . . 4 ⊢ (𝜑 → 𝑀 = ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩)
36	31, 35	opeq12d 4850	. . 3 ⊢ (𝜑 → ⟨𝑅, 𝑀⟩ = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
37	27, 36	eqtrd 2804	. 2 ⊢ (𝜑 → 𝑉 = ⟨⟨(1st ‘𝑅), (2nd ‘𝑅)⟩, ⟨(1st ‘𝑀), (2nd ‘𝑀)⟩⟩)
38	18, 17	nat1st2nd 18011	. 2 ⊢ (𝜑 → 𝐴 ∈ (⟨(1st ‘𝐹), (2nd ‘𝐹)⟩(𝐶 Nat 𝐷)⟨(1st ‘𝑀), (2nd ‘𝑀)⟩))
39	7, 6	nat1st2nd 18011	. 2 ⊢ (𝜑 → 𝐵 ∈ (⟨(1st ‘𝐾), (2nd ‘𝐾)⟩(𝐷 Nat 𝐸)⟨(1st ‘𝑅), (2nd ‘𝑅)⟩))
40	1, 26, 37, 38, 39	fuco22 50036	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 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  ⟨cop 4600   ↦ cmpt 5196   × cxp 5660  Rel wrel 5667  ‘cfv 6537  (class class class)co 7411  1^st c1st 7984  2^nd c2nd 7985  Basecbs 17269  compcco 17322   Func cfunc 17911   Nat cnat 18001   ∘_F cfuco 50013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-ixp 8896  df-func 17915  df-cofu 17917  df-nat 18003  df-fuco 50014

This theorem is referenced by:  fucocolem2  50051  fucocolem4  50053  fucolid  50058  fucorid  50059  precofvalALT  50065