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Theorem bj-idres 36672
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 6056 (see idinxpres 6055). See also elrid 6054 and elidinxp 6052. (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 5694 . 2 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2 inss1 4231 . . . 4 ( I ∩ (𝐴 × V)) ⊆ I
3 relinxp 5820 . . . . 5 Rel ( I ∩ (𝐴 × V))
4 elin 3965 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5 bj-opelidb1 36665 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
65simprbi 495 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7 opelxp1 5724 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥𝐴)
8 simpr 483 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
9 eleq1w 2812 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
109biimpa 475 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
118, 10jca 510 . . . . . . . 8 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
126, 7, 11syl2an 594 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
134, 12sylbi 216 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
14 opelxpi 5719 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
1513, 14syl 17 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
163, 15relssi 5793 . . . 4 ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
172, 16ssini 4234 . . 3 ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18 ssv 4006 . . . 4 𝐴 ⊆ V
19 xpss2 5702 . . . 4 (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20 sslin 4237 . . . 4 ((𝐴 × 𝐴) ⊆ (𝐴 × V) → ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V)))
2118, 19, 20mp2b 10 . . 3 ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V))
2217, 21eqssi 3998 . 2 ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
231, 22eqtri 2756 1 ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  Vcvv 3473  cin 3948  wss 3949  cop 4638   I cid 5579   × cxp 5680  cres 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-res 5694
This theorem is referenced by:  bj-opelidres  36673
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