Theorem cofu1st2nd 50020

Description: Rewrite the functor composition with separated functor parts. (Contributed by Zhi Wang, 15-Nov-2025.)

Hypotheses

Ref	Expression
cofu1st2nd.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
cofu1st2nd.g	⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))

Assertion

Ref	Expression
cofu1st2nd	⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))

Proof of Theorem cofu1st2nd

Step	Hyp	Ref	Expression
1		relfunc 17955	. . 3 ⊢ Rel (𝐷 Func 𝐸)
2		cofu1st2nd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
3		1st2nd 8039	. . 3 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
4	1, 2, 3	sylancr 599	. 2 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
5		relfunc 17955	. . 3 ⊢ Rel (𝐶 Func 𝐷)
6		cofu1st2nd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		1st2nd 8039	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
8	5, 6, 7	sylancr 599	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
9	4, 8	oveq12d 7434	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4593  Rel wrel 5664  ‘cfv 6537  (class class class)co 7416  1^st c1st 7987  2^nd c2nd 7988   Func cfunc 17947   ∘_func ccofu 17949

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7989  df-2nd 7990  df-func 17951

This theorem is used by:  uptrlem2  50139  uptra  50143  uobeqw  50147  uobeq  50148  uptr2a  50150