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Theorem bj-idres 35681
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 6005 (see idinxpres 6004). See also elrid 6003 and elidinxp 6001. (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 5649 . 2 ( I ↾ 𝐴) = ( I ∩ (𝐴 × V))
2 inss1 4192 . . . 4 ( I ∩ (𝐴 × V)) ⊆ I
3 relinxp 5774 . . . . 5 Rel ( I ∩ (𝐴 × V))
4 elin 3930 . . . . . . 7 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) ↔ (⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)))
5 bj-opelidb1 35674 . . . . . . . . 9 (⟨𝑥, 𝑦⟩ ∈ I ↔ (𝑥 ∈ V ∧ 𝑥 = 𝑦))
65simprbi 498 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ I → 𝑥 = 𝑦)
7 opelxp1 5678 . . . . . . . 8 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × V) → 𝑥𝐴)
8 simpr 486 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑥𝐴)
9 eleq1w 2817 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
109biimpa 478 . . . . . . . . 9 ((𝑥 = 𝑦𝑥𝐴) → 𝑦𝐴)
118, 10jca 513 . . . . . . . 8 ((𝑥 = 𝑦𝑥𝐴) → (𝑥𝐴𝑦𝐴))
126, 7, 11syl2an 597 . . . . . . 7 ((⟨𝑥, 𝑦⟩ ∈ I ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
134, 12sylbi 216 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → (𝑥𝐴𝑦𝐴))
14 opelxpi 5674 . . . . . 6 ((𝑥𝐴𝑦𝐴) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
1513, 14syl 17 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ ( I ∩ (𝐴 × V)) → ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐴))
163, 15relssi 5747 . . . 4 ( I ∩ (𝐴 × V)) ⊆ (𝐴 × 𝐴)
172, 16ssini 4195 . . 3 ( I ∩ (𝐴 × V)) ⊆ ( I ∩ (𝐴 × 𝐴))
18 ssv 3972 . . . 4 𝐴 ⊆ V
19 xpss2 5657 . . . 4 (𝐴 ⊆ V → (𝐴 × 𝐴) ⊆ (𝐴 × V))
20 sslin 4198 . . . 4 ((𝐴 × 𝐴) ⊆ (𝐴 × V) → ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V)))
2118, 19, 20mp2b 10 . . 3 ( I ∩ (𝐴 × 𝐴)) ⊆ ( I ∩ (𝐴 × V))
2217, 21eqssi 3964 . 2 ( I ∩ (𝐴 × V)) = ( I ∩ (𝐴 × 𝐴))
231, 22eqtri 2761 1 ( I ↾ 𝐴) = ( I ∩ (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  Vcvv 3447  cin 3913  wss 3914  cop 4596   I cid 5534   × cxp 5635  cres 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-res 5649
This theorem is referenced by:  bj-opelidres  35682
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