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Theorem idreseqidinxp 37482
Description: Condition for the identity restriction to be equal to the identity intersection with a Cartesian product. (Contributed by Peter Mazsa, 19-Jul-2018.)
Assertion
Ref Expression
idreseqidinxp (𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))

Proof of Theorem idreseqidinxp
StepHypRef Expression
1 inxpssres 5694 . . 3 ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴)
21a1i 11 . 2 (𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴))
3 idresssidinxp 37481 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
42, 3eqssd 4000 1 (𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3948  wss 3949   I cid 5574   × cxp 5675  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-res 5689
This theorem is referenced by:  symrefref2  37737
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