Theorem fucofn22 49827

Description: The morphism part of the functor composition bifunctor maps two natural transformations to a function on a base set. (Contributed by Zhi Wang, 30-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
fucofn22	⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) Fn (Base‘𝐶))

Proof of Theorem fucofn22

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ovex 7393	. . 3 ⊢ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥))) ∈ V
2		eqid 2737	. . 3 ⊢ (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))))
3	1, 2	fnmpti 6635	. 2 ⊢ (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))) Fn (Base‘𝐶)
4		fuco22.o	. . . 4 ⊢ (𝜑 → (⟨𝐶, 𝐷⟩ ∘F 𝐸) = ⟨𝑂, 𝑃⟩)
5		fuco22.u	. . . 4 ⊢ (𝜑 → 𝑈 = ⟨⟨𝐾, 𝐿⟩, ⟨𝐹, 𝐺⟩⟩)
6		fuco22.v	. . . 4 ⊢ (𝜑 → 𝑉 = ⟨⟨𝑅, 𝑆⟩, ⟨𝑀, 𝑁⟩⟩)
7		fuco22.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ (⟨𝐹, 𝐺⟩(𝐶 Nat 𝐷)⟨𝑀, 𝑁⟩))
8		fuco22.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ (⟨𝐾, 𝐿⟩(𝐷 Nat 𝐸)⟨𝑅, 𝑆⟩))
9	4, 5, 6, 7, 8	fuco22 49826	. . 3 ⊢ (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)))))
10	9	fneq1d 6585	. 2 ⊢ (𝜑 → ((𝐵(𝑈𝑃𝑉)𝐴) Fn (Base‘𝐶) ↔ (𝑥 ∈ (Base‘𝐶) ↦ ((𝐵‘(𝑀‘𝑥))(⟨(𝐾‘(𝐹‘𝑥)), (𝐾‘(𝑀‘𝑥))⟩(comp‘𝐸)(𝑅‘(𝑀‘𝑥)))(((𝐹‘𝑥)𝐿(𝑀‘𝑥))‘(𝐴‘𝑥)))) Fn (Base‘𝐶)))
11	3, 10	mpbiri 258	1 ⊢ (𝜑 → (𝐵(𝑈𝑃𝑉)𝐴) Fn (Base‘𝐶))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4574   ↦ cmpt 5167   Fn wfn 6487  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  compcco 17223   Nat cnat 17902   ∘_F cfuco 49803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-ixp 8839  df-func 17816  df-cofu 17818  df-nat 17904  df-fuco 49804

This theorem is referenced by:  fuco22natlem  49832