Theorem idreseqidinxp 38357

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 5631	. . 3 ⊢ ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴)
2	1	a1i 11	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴))
3		idresssidinxp 38356	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
4	2, 3	eqssd 3947	1 ⊢ (𝐴 ⊆ 𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3896   ⊆ wss 3897   I cid 5508   × cxp 5612   ↾ cres 5616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-res 5626

This theorem is referenced by:  symrefref2  38669