Theorem cofu1st2nd 49525

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 17787	. . 3 ⊢ Rel (𝐷 Func 𝐸)
2		cofu1st2nd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
3		1st2nd 7983	. . 3 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐺 ∈ (𝐷 Func 𝐸)) → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
4	1, 2, 3	sylancr 588	. 2 ⊢ (𝜑 → 𝐺 = ⟨(1st ‘𝐺), (2nd ‘𝐺)⟩)
5		relfunc 17787	. . 3 ⊢ Rel (𝐶 Func 𝐷)
6		cofu1st2nd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		1st2nd 7983	. . 3 ⊢ ((Rel (𝐶 Func 𝐷) ∧ 𝐹 ∈ (𝐶 Func 𝐷)) → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
8	5, 6, 7	sylancr 588	. 2 ⊢ (𝜑 → 𝐹 = ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩)
9	4, 8	oveq12d 7376	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4574  Rel wrel 5627  ‘cfv 6490  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932   Func cfunc 17779   ∘_func ccofu 17781

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-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

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-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-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 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-func 17783

This theorem is referenced by:  uptrlem2  49644  uptra  49648  uobeqw  49652  uobeq  49653  uptr2a  49655