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Theorem idresssidinxp 37832
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 6002 . . 3 ( I ↾ 𝐴) ⊆ I
21a1i 11 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ I )
3 idssxp 6048 . . 3 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
4 xpss2 5693 . . 3 (𝐴𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵))
53, 4sstrid 3985 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵))
62, 5ssind 4228 1 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3940  wss 3941   I cid 5570   × cxp 5671  cres 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-id 5571  df-xp 5679  df-rel 5680  df-res 5685
This theorem is referenced by:  idreseqidinxp  37833
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