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Theorem ideqg 5195
 Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
ideqg (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))

Proof of Theorem ideqg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 22 . . 3 (𝐵𝑉𝐵𝑉)
2 reli 5171 . . . 4 Rel I
32brrelexi 5082 . . 3 (𝐴 I 𝐵𝐴 ∈ V)
41, 3anim12ci 589 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
5 eleq1 2676 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
65biimparc 503 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
76elexd 3187 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
8 simpl 472 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
97, 8jca 553 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 eqeq1 2614 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqeq2 2621 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
12 df-id 4953 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1310, 11, 12brabg 4919 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
144, 9, 13pm5.21nd 939 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583   I cid 4948 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045 This theorem is referenced by:  ideq  5196  ididg  5197  restidsingOLD  5378  poleloe  5446  isof1oidb  6474  pltval  16783  tglngne  25245  tgelrnln  25325  opeldifid  28794  fourierdlem42  39042
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