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Theorem idresssidinxp 37780
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 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))

Proof of Theorem idresssidinxp
StepHypRef Expression
1 resss 6010 . . 3 ( I ↾ 𝐴) ⊆ I
21a1i 11 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ I )
3 idssxp 6052 . . 3 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
4 xpss2 5698 . . 3 (𝐴𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵))
53, 4sstrid 3991 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵))
62, 5ssind 4233 1 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3946  wss 3947   I cid 5575   × cxp 5676  cres 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-res 5690
This theorem is referenced by:  idreseqidinxp  37781
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