Theorem fuco22 50555

Description: The morphism part of the functor composition bifunctor. See also fuco22a 50566. (Contributed by Zhi Wang, 29-Sep-2025.)

Hypotheses

Ref	Expression
fuco22.o	⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
fuco22.u	⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
fuco22.v	⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
fuco22.a	⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
fuco22.b	⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))

Assertion

Ref	Expression
fuco22	⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))))

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

Allowed substitution hints:   𝑃(𝑥)   𝑆(𝑥)   𝐺(𝑥)   𝑁(𝑥)   𝑂(𝑥)

Proof of Theorem fuco22

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fuco22.o	. . 3 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
2		eqid 2760	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
4	2, 3	natrcl2 50434	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
5		eqid 2760	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
7	5, 6	natrcl2 50434	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
8		fuco22.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
9	2, 3	natrcl3 50435	. . 3 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
10	5, 6	natrcl3 50435	. . 3 ⊢ (𝜑 → 𝑅(𝐷 Func 𝐸)𝑆)
11		fuco22.v	. . 3 ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
12	1, 4, 7, 8, 9, 10, 11	fuco21 50552	. 2 ⊢ (𝜑 → (𝑈𝑃𝑉) = (𝑏 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩), 𝑎 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩) ↦ (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝑎‘𝑥))))))
13		simplrl 789	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑏 = 𝐵)
14	13	fveq1d 6888	. . . 4 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑏‘(𝑀‘𝑥)) = (𝐵‘(𝑀‘𝑥)))
15		simplrr 790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑎 = 𝐴)
16	15	fveq1d 6888	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑎‘𝑥) = (𝐴‘𝑥))
17	16	fveq2d 6890	. . . 4 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝑎‘𝑥)) = (((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))
18	14, 17	oveq12d 7439	. . 3 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → ((𝑏‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝑎‘𝑥))) = ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥))))
19	18	mpteq2dva 5197	. 2 ⊢ ((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) → (𝑥 ∈ (Base‘𝐶) ↦ ((𝑏‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝑎‘𝑥)))) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))))
20		fvexd 6901	. . 3 ⊢ (𝜑 → (Base‘𝐶) ∈ V)
21	20	mptexd 7231	. 2 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))) ∈ V)
22	12, 19, 6, 3, 21	ovmpod 7573	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) = (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⟨cop 4834   ↦ cmpt 5185  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  compcco 17446   Nat cnat 18125   ∘_F cfuco 50532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-ixp 8930  df-func 18039  df-cofu 18041  df-nat 18127  df-fuco 50533

This theorem is used by:  fucofn22  50556  fuco23  50557  fucof21  50563  fucoid  50564  fuco22a  50566