Theorem fuco22 50100

Description: The morphism part of the functor composition bifunctor. See also fuco22a 50111. (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 2763	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
4	2, 3	natrcl2 49985	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
5		eqid 2763	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
7	5, 6	natrcl2 49985	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
8		fuco22.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
9	2, 3	natrcl3 49986	. . 3 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
10	5, 6	natrcl3 49986	. . 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 50097	. 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 788	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑏 = 𝐵)
14	13	fveq1d 6885	. . . 4 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑏‘(𝑀‘𝑥)) = (𝐵‘(𝑀‘𝑥)))
15		simplrr 789	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑎 = 𝐴)
16	15	fveq1d 6885	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑎‘𝑥) = (𝐴‘𝑥))
17	16	fveq2d 6887	. . . 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 7430	. . 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 5205	. 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 6898	. . 3 ⊢ (𝜑 → (Base‘𝐶) ∈ V)
21	20	mptexd 7224	. 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 7564	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)))))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ⟨cop 4596   ↦ cmpt 5193  ‘cfv 6538  (class class class)co 7412  Basecbs 17270  compcco 17323   Nat cnat 18002   ∘_F cfuco 50077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-ixp 8897  df-func 17916  df-cofu 17918  df-nat 18004  df-fuco 50078

This theorem is referenced by:  fucofn22  50101  fuco23  50102  fucof21  50108  fucoid  50109  fuco22a  50111