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Theorem bj-idres 37143
Description: Alternate expression for the restricted identity relation. The advantage of that expression is to expose it as a "bounded" class, being included in the Cartesian square of the restricting class. (Contributed by BJ, 27-Dec-2023.)

This is an alternate of idinxpresid 6068 (see idinxpres 6067). See also elrid 6066 and elidinxp 6064. (Proof modification is discouraged.)

Assertion
Ref Expression
bj-idres ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))

Proof of Theorem bj-idres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-res 5701 . 2 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2 inss1 4245 . . . 4 ( I ∩ (𝐴 × V)) ⊆ I
3 relinxp 5827 . . . . 5 Rel ( I ∩ (𝐴 × V))
4 elin 3979 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5 bj-opelidb1 37136 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
65simprbi 496 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7 opelxp1 5731 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥𝐴)
8 simpr 484 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
9 eleq1w 2822 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
109biimpa 476 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
118, 10jca 511 . . . . . . . 8 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
126, 7, 11syl2an 596 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
134, 12sylbi 217 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
14 opelxpi 5726 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
1513, 14syl 17 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
163, 15relssi 5800 . . . 4 ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
172, 16ssini 4248 . . 3 ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18 ssv 4020 . . . 4 𝐴 ⊆ V
19 xpss2 5709 . . . 4 (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20 sslin 4251 . . . 4 ((𝐴 × 𝐴) ⊆ (𝐴 × V) → ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V)))
2118, 19, 20mp2b 10 . . 3 ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V))
2217, 21eqssi 4012 . 2 ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
231, 22eqtri 2763 1 ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963  cop 4637   I cid 5582   × cxp 5687  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by:  bj-opelidres  37144
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