Metamath Proof Explorer


Theorem idinxpssinxp2

Description: Identity intersection with a square Cartesian product in subclass relation with an intersection with the same Cartesian product. (Contributed by Peter Mazsa, 4-Mar-2019) (Proof modification is discouraged.)

Ref Expression
Assertion idinxpssinxp2 ( ( I ∩ ( 𝐴 × 𝐴 ) ) ⊆ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) ↔ ∀ 𝑥𝐴 𝑥 𝑅 𝑥 )

Proof

Step Hyp Ref Expression
1 idinxpresid ( I ∩ ( 𝐴 × 𝐴 ) ) = ( I ↾ 𝐴 )
2 1 sseq1i ( ( I ∩ ( 𝐴 × 𝐴 ) ) ⊆ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) ↔ ( I ↾ 𝐴 ) ⊆ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) )
3 idrefALT ( ( I ↾ 𝐴 ) ⊆ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) ↔ ∀ 𝑥𝐴 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 )
4 brinxp2 ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ↔ ( ( 𝑥𝐴𝑥𝐴 ) ∧ 𝑥 𝑅 𝑥 ) )
5 pm4.24 ( 𝑥𝐴 ↔ ( 𝑥𝐴𝑥𝐴 ) )
6 5 anbi1i ( ( 𝑥𝐴𝑥 𝑅 𝑥 ) ↔ ( ( 𝑥𝐴𝑥𝐴 ) ∧ 𝑥 𝑅 𝑥 ) )
7 4 6 bitr4i ( 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ↔ ( 𝑥𝐴𝑥 𝑅 𝑥 ) )
8 7 ralbii ( ∀ 𝑥𝐴 𝑥 ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) 𝑥 ↔ ∀ 𝑥𝐴 ( 𝑥𝐴𝑥 𝑅 𝑥 ) )
9 2 3 8 3bitri ( ( I ∩ ( 𝐴 × 𝐴 ) ) ⊆ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) ↔ ∀ 𝑥𝐴 ( 𝑥𝐴𝑥 𝑅 𝑥 ) )
10 ralanid ( ∀ 𝑥𝐴 ( 𝑥𝐴𝑥 𝑅 𝑥 ) ↔ ∀ 𝑥𝐴 𝑥 𝑅 𝑥 )
11 9 10 bitri ( ( I ∩ ( 𝐴 × 𝐴 ) ) ⊆ ( 𝑅 ∩ ( 𝐴 × 𝐴 ) ) ↔ ∀ 𝑥𝐴 𝑥 𝑅 𝑥 )