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Metamath Proof Explorer

Theorem cossssid5

Description: Equivalent expressions for the class of cosets by R to be a subset of the identity class. (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion cossssid5 ⊢ ≀ R ⊆ I ↔ ∀ x ∈ ran ⁡ R ∀ y ∈ ran ⁡ R x = y ∨ x R -1 ∩ y R -1 = ∅

Proof

Step Hyp Ref Expression
1 cossssid4 ⊢ ≀ R ⊆ I ↔ ∀ u ∃* x u R x
2 ineccnvmo2 ⊢ ∀ x ∈ ran ⁡ R ∀ y ∈ ran ⁡ R x = y ∨ x R -1 ∩ y R -1 = ∅ ↔ ∀ u ∃* x u R x
3 1 2 bitr4i ⊢ ≀ R ⊆ I ↔ ∀ x ∈ ran ⁡ R ∀ y ∈ ran ⁡ R x = y ∨ x R -1 ∩ y R -1 = ∅