Theorem idreseqidinxp 38310

Description: Condition for the identity restriction to be equal to the identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.)

Assertion

Ref	Expression
idreseqidinxp	⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))

Proof of Theorem idreseqidinxp

Step	Hyp	Ref	Expression
1		inxpssres 5702	. . 3 ⊢ ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴)
2	1	a1i 11	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴))
3		idresssidinxp 38309	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
4	2, 3	eqssd 4001	1 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3950   ⊆ wss 3951   I cid 5577   × cxp 5683   ↾ cres 5687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-res 5697

This theorem is referenced by:  symrefref2  38564