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Theorem bj-idreseq 37145
Description: Sufficient condition for the restricted identity relation to agree with equality. Note that the instance of bj-ideqg 37140 with V substituted for 𝑉 is a direct consequence of bj-idreseq 37145. This is a strengthening of resieq 6011 which should be proved from it (note that currently, resieq 6011 relies on ideq 5866). Note that the intersection in the antecedent is not very meaningful, but is a device to prove versions with either class assumed to be a set. It could be enough to prove the version with a disjunctive antecedent: ((𝐴𝐶𝐵𝐶) → ...). (Contributed by BJ, 25-Dec-2023.)
Assertion
Ref Expression
bj-idreseq ((𝐴𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵𝐴 = 𝐵))

Proof of Theorem bj-idreseq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-brresdm 37129 . . . 4 (𝐴( I ↾ 𝐶)𝐵𝐴𝐶)
2 relres 6026 . . . . 5 Rel ( I ↾ 𝐶)
32brrelex2i 5746 . . . 4 (𝐴( I ↾ 𝐶)𝐵𝐵 ∈ V)
41, 3jca 511 . . 3 (𝐴( I ↾ 𝐶)𝐵 → (𝐴𝐶𝐵 ∈ V))
54adantl 481 . 2 (((𝐴𝐵) ∈ 𝐶𝐴( I ↾ 𝐶)𝐵) → (𝐴𝐶𝐵 ∈ V))
6 eqimss 4054 . . . . . 6 (𝐴 = 𝐵𝐴𝐵)
7 dfss2 3981 . . . . . 6 (𝐴𝐵 ↔ (𝐴𝐵) = 𝐴)
86, 7sylib 218 . . . . 5 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐴)
98adantl 481 . . . 4 (((𝐴𝐵) ∈ 𝐶𝐴 = 𝐵) → (𝐴𝐵) = 𝐴)
10 simpl 482 . . . 4 (((𝐴𝐵) ∈ 𝐶𝐴 = 𝐵) → (𝐴𝐵) ∈ 𝐶)
119, 10eqeltrrd 2840 . . 3 (((𝐴𝐵) ∈ 𝐶𝐴 = 𝐵) → 𝐴𝐶)
12 eqimss2 4055 . . . . . . 7 (𝐴 = 𝐵𝐵𝐴)
13 sseqin2 4231 . . . . . . 7 (𝐵𝐴 ↔ (𝐴𝐵) = 𝐵)
1412, 13sylib 218 . . . . . 6 (𝐴 = 𝐵 → (𝐴𝐵) = 𝐵)
1514adantl 481 . . . . 5 (((𝐴𝐵) ∈ 𝐶𝐴 = 𝐵) → (𝐴𝐵) = 𝐵)
1615, 10eqeltrrd 2840 . . . 4 (((𝐴𝐵) ∈ 𝐶𝐴 = 𝐵) → 𝐵𝐶)
1716elexd 3502 . . 3 (((𝐴𝐵) ∈ 𝐶𝐴 = 𝐵) → 𝐵 ∈ V)
1811, 17jca 511 . 2 (((𝐴𝐵) ∈ 𝐶𝐴 = 𝐵) → (𝐴𝐶𝐵 ∈ V))
19 brres 6007 . . . 4 (𝐵 ∈ V → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 I 𝐵)))
2019adantl 481 . . 3 ((𝐴𝐶𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵 ↔ (𝐴𝐶𝐴 I 𝐵)))
21 eqeq12 2752 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝑥 = 𝑦𝐴 = 𝐵))
22 df-id 5583 . . . . 5 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
2321, 22brabga 5544 . . . 4 ((𝐴𝐶𝐵 ∈ V) → (𝐴 I 𝐵𝐴 = 𝐵))
2423anbi2d 630 . . 3 ((𝐴𝐶𝐵 ∈ V) → ((𝐴𝐶𝐴 I 𝐵) ↔ (𝐴𝐶𝐴 = 𝐵)))
25 simp3 1137 . . . . 5 (((𝐴𝐶𝐵 ∈ V) ∧ 𝐴𝐶𝐴 = 𝐵) → 𝐴 = 𝐵)
26253expib 1121 . . . 4 ((𝐴𝐶𝐵 ∈ V) → ((𝐴𝐶𝐴 = 𝐵) → 𝐴 = 𝐵))
27 3simpb 1148 . . . . 5 ((𝐴𝐶𝐵 ∈ V ∧ 𝐴 = 𝐵) → (𝐴𝐶𝐴 = 𝐵))
28273expia 1120 . . . 4 ((𝐴𝐶𝐵 ∈ V) → (𝐴 = 𝐵 → (𝐴𝐶𝐴 = 𝐵)))
2926, 28impbid 212 . . 3 ((𝐴𝐶𝐵 ∈ V) → ((𝐴𝐶𝐴 = 𝐵) ↔ 𝐴 = 𝐵))
3020, 24, 293bitrd 305 . 2 ((𝐴𝐶𝐵 ∈ V) → (𝐴( I ↾ 𝐶)𝐵𝐴 = 𝐵))
315, 18, 30pm5.21nd 802 1 ((𝐴𝐵) ∈ 𝐶 → (𝐴( I ↾ 𝐶)𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963   class class class wbr 5148   I cid 5582  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by: (None)
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