Metamath Proof Explorer

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