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Theorem bj-idres 34456
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 5918 (see idinxpres 5917). See also elrid 5916 and elidinxp 5914. (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 5570 . 2 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2 inss1 4208 . . . 4 ( I ∩ (𝐴 × V)) ⊆ I
3 relinxp 5690 . . . . 5 Rel ( I ∩ (𝐴 × V))
4 elin 4172 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5 bj-opelidb1 34449 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
65simprbi 499 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7 opelxp1 5599 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥𝐴)
8 simpr 487 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
9 eleq1w 2898 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
109biimpa 479 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
118, 10jca 514 . . . . . . . 8 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
126, 7, 11syl2an 597 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
134, 12sylbi 219 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
14 opelxpi 5595 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
1513, 14syl 17 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
163, 15relssi 5663 . . . 4 ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
172, 16ssini 4211 . . 3 ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18 ssv 3994 . . . 4 𝐴 ⊆ V
19 xpss2 5578 . . . 4 (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20 sslin 4214 . . . 4 ((𝐴 × 𝐴) ⊆ (𝐴 × V) → ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V)))
2118, 19, 20mp2b 10 . . 3 ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V))
2217, 21eqssi 3986 . 2 ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
231, 22eqtri 2847 1 ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1536  wcel 2113  Vcvv 3497  cin 3938  wss 3939  cop 4576   I cid 5462   × cxp 5556  cres 5560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pr 5333
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-opab 5132  df-id 5463  df-xp 5564  df-rel 5565  df-res 5570
This theorem is referenced by:  bj-opelidres  34457
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