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Theorem idreseqidinxp 35743
 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 5536 . . 3 ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴)
21a1i 11 . 2 (𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) ⊆ ( I ↾ 𝐴))
3 idresssidinxp 35742 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
42, 3eqssd 3932 1 (𝐴𝐵 → ( I ∩ (𝐴 × 𝐵)) = ( I ↾ 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∩ cin 3880   ⊆ wss 3881   I cid 5424   × cxp 5517   ↾ cres 5521 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-res 5531 This theorem is referenced by:  symrefref2  35975
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