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Theorem idresssidinxp 34621
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 5658 . . 3 ( I ↾ 𝐴) ⊆ I
21a1i 11 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ I )
3 idssxp 5697 . . 3 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
4 xpss2 5362 . . 3 (𝐴𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵))
53, 4syl5ss 3838 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵))
62, 5ssind 4061 1 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3797  wss 3798   I cid 5249   × cxp 5340  cres 5344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-res 5354
This theorem is referenced by:  idreseqidinxp  34622
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