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Metamath Proof Explorer

Theorem extssr

Description: Property of subset relation, see also extid , extep and the comment of df-ssr . (Contributed by Peter Mazsa, 10-Jul-2019)

Ref Expression
Assertion extssr ⊢ A ∈ V ∧ B ∈ W → A S -1 = B S -1 ↔ A = B

Proof

Step Hyp Ref Expression
1 brssr ⊢ A ∈ V → x S A ↔ x ⊆ A
2 brssr ⊢ B ∈ W → x S B ↔ x ⊆ B
3 1 2 bi2bian9 ⊢ A ∈ V ∧ B ∈ W → x S A ↔ x S B ↔ x ⊆ A ↔ x ⊆ B
4 3 albidv ⊢ A ∈ V ∧ B ∈ W → ∀ x x S A ↔ x S B ↔ ∀ x x ⊆ A ↔ x ⊆ B
5 relssr ⊢ Rel ⁡ S
6 releccnveq ⊢ Rel ⁡ S ∧ Rel ⁡ S → A S -1 = B S -1 ↔ ∀ x x S A ↔ x S B
7 5 5 6 mp2an ⊢ A S -1 = B S -1 ↔ ∀ x x S A ↔ x S B
8 ssext ⊢ A = B ↔ ∀ x x ⊆ A ↔ x ⊆ B
9 4 7 8 3bitr4g ⊢ A ∈ V ∧ B ∈ W → A S -1 = B S -1 ↔ A = B