Theorem cofu1st2nd 49567

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 17829	. . 3 ⊢ Rel (𝐷 Func 𝐸)
2		cofu1st2nd.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐷 Func 𝐸))
3		1st2nd 7992	. . 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 17829	. . 3 ⊢ Rel (𝐶 Func 𝐷)
6		cofu1st2nd.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
7		1st2nd 7992	. . 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 7385	1 ⊢ (𝜑 → (𝐺 ∘func 𝐹) = (⟨(1st ‘𝐺), (2nd ‘𝐺)⟩ ∘func ⟨(1st ‘𝐹), (2nd ‘𝐹)⟩))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4573  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941   Func cfunc 17821   ∘_func ccofu 17823

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 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-func 17825

This theorem is referenced by:  uptrlem2  49686  uptra  49690  uobeqw  49694  uobeq  49695  uptr2a  49697