Theorem cotrgOLDOLD

Description: Obsolete version of cotrg as of 19-Dec-2024. (Contributed by NM, 27-Dec-1996) (Proof shortened by Andrew Salmon, 27-Aug-2011) Generalized from its special instance cotr . (Revised by Richard Penner, 24-Dec-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	cotrgOLDOLD	⊢ ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )

Proof

Step	Hyp	Ref	Expression
1		relco	⊢ Rel ( 𝐴 ∘ 𝐵 )
2		ssrel	⊢ ( Rel ( 𝐴 ∘ 𝐵 ) → ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑧 ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) → ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑧 ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) → ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ) )
4		vex	⊢ 𝑥 ∈ V
5		vex	⊢ 𝑧 ∈ V
6	4 5	opelco	⊢ ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) ↔ ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) )
7		df-br	⊢ ( 𝑥 𝐶 𝑧 ↔ ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 )
8	7	bicomi	⊢ ( ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ↔ 𝑥 𝐶 𝑧 )
9	6 8	imbi12i	⊢ ( ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) → ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ) ↔ ( ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
10		19.23v	⊢ ( ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ( ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
11	9 10	bitr4i	⊢ ( ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) → ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ) ↔ ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
12	11	albii	⊢ ( ∀ 𝑧 ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) → ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ) ↔ ∀ 𝑧 ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
13		alcom	⊢ ( ∀ 𝑧 ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
14	12 13	bitri	⊢ ( ∀ 𝑧 ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) → ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
15	14	albii	⊢ ( ∀ 𝑥 ∀ 𝑧 ( ⟨ 𝑥 , 𝑧 ⟩ ∈ ( 𝐴 ∘ 𝐵 ) → ⟨ 𝑥 , 𝑧 ⟩ ∈ 𝐶 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
16	3 15	bitri	⊢ ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )