Theorem fuco22 48994

Description: The morphism part of the functor composition bifunctor. See also fuco22a 49005. (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 2734	. . . 4 ⊢ (𝐶 Nat 𝐷) = (𝐶 Nat 𝐷)
3		fuco22.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
4	2, 3	natrcl2 48905	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
5		eqid 2734	. . . 4 ⊢ (𝐷 Nat 𝐸) = (𝐷 Nat 𝐸)
6		fuco22.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
7	5, 6	natrcl2 48905	. . 3 ⊢ (𝜑 → 𝐾(𝐷 Func 𝐸)𝐿)
8		fuco22.u	. . 3 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
9	2, 3	natrcl3 48906	. . 3 ⊢ (𝜑 → 𝑀(𝐶 Func 𝐷)𝑁)
10	5, 6	natrcl3 48906	. . 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 48991	. 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 6889	. . . 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 6889	. . . . 5 ⊢ (((𝜑 ∧ (𝑏 = 𝐵 ∧ 𝑎 = 𝐴)) ∧ 𝑥 ∈ (Base‘𝐶)) → (𝑎‘𝑥) = (𝐴‘𝑥))
17	16	fveq2d 6891	. . . 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 7432	. . 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 5224	. 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 6902	. . 3 ⊢ (𝜑 → (Base‘𝐶) ∈ V)
21	20	mptexd 7227	. 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 7568	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 1539   ∈ wcel 2107  Vcvv 3464  ⟨cop 4614   ↦ cmpt 5207  ‘cfv 6542  (class class class)co 7414  Basecbs 17230  compcco 17286   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:  fucofn22  48995  fuco23  48996  fucof21  49002  fucoid  49003  fuco22a  49005