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Theorem idresssidinxp 35568
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 5880 . . 3 ( I ↾ 𝐴) ⊆ I
21a1i 11 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ I )
3 idssxp 5918 . . 3 ( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)
4 xpss2 5577 . . 3 (𝐴𝐵 → (𝐴 × 𝐴) ⊆ (𝐴 × 𝐵))
53, 4sstrid 3980 . 2 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ (𝐴 × 𝐵))
62, 5ssind 4211 1 (𝐴𝐵 → ( I ↾ 𝐴) ⊆ ( I ∩ (𝐴 × 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3937  wss 3938   I cid 5461   × cxp 5555  cres 5559
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-res 5569
This theorem is referenced by:  idreseqidinxp  35569
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