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Theorem bj-idreseqb 34493
 Description: Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023.)
Assertion
Ref Expression
bj-idreseqb (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 = 𝐵))

Proof of Theorem bj-idreseqb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5870 . . 3 Rel ( I ↾ 𝐶)
21brrelex12i 5595 . 2 (𝐴( I ↾ 𝐶)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3 simpl 486 . . . 4 ((𝐴𝐶𝐴 = 𝐵) → 𝐴𝐶)
43elexd 3500 . . 3 ((𝐴𝐶𝐴 = 𝐵) → 𝐴 ∈ V)
5 eleq1 2903 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐶𝐵𝐶))
65biimpac 482 . . . 4 ((𝐴𝐶𝐴 = 𝐵) → 𝐵𝐶)
76elexd 3500 . . 3 ((𝐴𝐶𝐴 = 𝐵) → 𝐵 ∈ V)
84, 7jca 515 . 2 ((𝐴𝐶𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
9 brres 5848 . . . 4 (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 I 𝐵)))
109adantl 485 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 I 𝐵)))
11 eqeq12 2838 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
12 df-id 5448 . . . . 5 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1311, 12brabga 5409 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴 I 𝐵𝐴 = 𝐵))
1413anbi2d 631 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝐴𝐶𝐴 I 𝐵) ↔ (𝐴𝐶𝐴 = 𝐵)))
1510, 14bitrd 282 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 = 𝐵)))
162, 8, 15pm5.21nii 383 1 (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3480   class class class wbr 5053   I cid 5447   ↾ cres 5545 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-res 5555 This theorem is referenced by:  bj-elid7  34501
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