Metamath Proof Explorer

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