Theorem cotrgOLD

Description: Obsolete version of cotrg as of 29-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 shortened by SN, 19-Dec-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	cotrgOLD	⊢ ( ( 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		ssrel3	⊢ ( Rel ( 𝐴 ∘ 𝐵 ) → ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ) )
3	1 2	ax-mp	⊢ ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) )
4		vex	⊢ 𝑥 ∈ V
5		vex	⊢ 𝑧 ∈ V
6	4 5	brco	⊢ ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 ↔ ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) )
7	6	imbi1i	⊢ ( ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ( ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
8		19.23v	⊢ ( ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ( ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
9	7 8	bitr4i	⊢ ( ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
10	9	albii	⊢ ( ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑧 ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
11		alcom	⊢ ( ∀ 𝑧 ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
12	10 11	bitri	⊢ ( ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
13	12	albii	⊢ ( ∀ 𝑥 ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )
14	3 13	bitri	⊢ ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) )