Theorem fuco22 49464

Description: The morphism part of the functor composition bifunctor. See also fuco22a 49475. (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 2733	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
4	2, 3	natrcl2 49349	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
5		eqid 2733	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
7	5, 6	natrcl2 49349	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
8		fuco22.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
9	2, 3	natrcl3 49350	. . 3 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
10	5, 6	natrcl3 49350	. . 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 49461	. 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 776	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑏 = 𝐵)
14	13	fveq1d 6830	. . . 4 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑏‘(𝑀‘𝑥)) = (𝐵‘(𝑀‘𝑥)))
15		simplrr 777	. . . . . 6 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → 𝑎 = 𝐴)
16	15	fveq1d 6830	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑎‘𝑥) = (𝐴‘𝑥))
17	16	fveq2d 6832	. . . 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 7370	. . 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 5186	. 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 6843	. . 3 ⊢ (𝜑 → (Base‘𝐶) ∈ V)
21	20	mptexd 7164	. 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 7504	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 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437  ⟨cop 4581   ↦ cmpt 5174  ‘cfv 6486  (class class class)co 7352  Basecbs 17122  compcco 17175   Nat cnat 17853   ∘_F cfuco 49441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7927  df-2nd 7928  df-ixp 8828  df-func 17767  df-cofu 17769  df-nat 17855  df-fuco 49442

This theorem is referenced by:  fucofn22  49465  fuco23  49466  fucof21  49472  fucoid  49473  fuco22a  49475