Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  idresssidinxp Structured version   Visualization version   GIF version

Theorem idresssidinxp 37703
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 6004 . . 3 ( I ↾ 𝐴) ⊆ I
21a1i 11 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ I )
3 idssxp 6046 . . 3 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
4 xpss2 5692 . . 3 (𝐴𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵))
53, 4sstrid 3989 . 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 3943  wss 3944   I cid 5569   × cxp 5670  cres 5674
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 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
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 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-res 5684
This theorem is referenced by:  idreseqidinxp  37704
  Copyright terms: Public domain W3C validator