Theorem idresssidinxp 38688

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 5960	. . 3 ⊢ ( I ↾ 𝐴) ⊆ I
2	1	a1i 11	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ I )
3		idssxp 6008	. . 3 ⊢ ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
4		xpss2 5645	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵))
5	3, 4	sstrid 3933	. 2 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵))
6	2, 5	ssind 4176	1 ⊢ (𝐴 ⊆ 𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))

Syntax hints:   → wi 4   ∩ cin 3889   ⊆ wss 3890   I cid 5519   × cxp 5623   ↾ cres 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-res 5637

This theorem is referenced by:  idreseqidinxp  38689