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Theorem idreseqidinxp 37047
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 5687 . . 3 ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴)
21a1i 11 . 2 (𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴))
3 idresssidinxp 37046 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
42, 3eqssd 3996 1 (𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cin 3944  wss 3945   I cid 5567   × cxp 5668  cres 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pr 5421
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5143  df-opab 5205  df-id 5568  df-xp 5676  df-rel 5677  df-res 5682
This theorem is referenced by:  symrefref2  37302
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