Theorem idresssidinxp 39004

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

Assertion

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

Proof of Theorem idresssidinxp

Step	Hyp	Ref	Expression
1		resss 6005	. . 3 ⊢ ( I ↾ 𝐴) ⊆ I
2	1	a1i 11	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ I )
3		idssxp 6056	. . 3 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
4		xpss2 5686	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵))
5	3, 4	sstrid 3951	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵))
6	2, 5	ssind 4196	1 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))

This proof depends on syntax axioms:   → wi 4   ∩ cin 3907   ⊆ wss 3908   I cid 5560   × cxp 5664   ↾ cres 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-res 5678

This theorem is used by:  idreseqidinxp  39005