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Theorem bj-idres 37162
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 6065 (see idinxpres 6064). See also elrid 6063 and elidinxp 6061. (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 5696 . 2 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2 inss1 4236 . . . 4 ( I ∩ (𝐴 × V)) ⊆ I
3 relinxp 5823 . . . . 5 Rel ( I ∩ (𝐴 × V))
4 elin 3966 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5 bj-opelidb1 37155 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
65simprbi 496 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7 opelxp1 5726 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥𝐴)
8 simpr 484 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
9 eleq1w 2823 . . . . . . . . . 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 5721 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
1513, 14syl 17 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
163, 15relssi 5796 . . . 4 ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
172, 16ssini 4239 . . 3 ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18 ssv 4007 . . . 4 𝐴 ⊆ V
19 xpss2 5704 . . . 4 (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20 sslin 4242 . . . 4 ((𝐴 × 𝐴) ⊆ (𝐴 × V) → ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V)))
2118, 19, 20mp2b 10 . . 3 ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V))
2217, 21eqssi 3999 . 2 ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
231, 22eqtri 2764 1 ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1539  wcel 2107  Vcvv 3479  cin 3949  wss 3950  cop 4631   I cid 5576   × cxp 5682  cres 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-res 5696
This theorem is referenced by:  bj-opelidres  37163
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