MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ideqg Structured version   Visualization version   GIF version

Theorem ideqg 5306
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 5282 . . . 4 Rel I
32brrelexi 5192 . . 3 (𝐴 I 𝐵𝐴 ∈ V)
41, 3anim12ci 590 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
5 eleq1 2718 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
65biimparc 503 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
76elexd 3245 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
8 simpl 472 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
97, 8jca 553 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 eqeq1 2655 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqeq2 2662 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
12 df-id 5053 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1310, 11, 12brabg 5023 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
144, 9, 13pm5.21nd 961 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  Vcvv 3231   class class class wbr 4685   I cid 5052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150
This theorem is referenced by:  ideq  5307  ididg  5308  restidsingOLD  5494  poleloe  5562  isof1oidb  6614  pltval  17007  tglngne  25490  tgelrnln  25570  opeldifid  29538  ideq2  34219  idinxpss  34224  inxpssidinxp  34227  idinxpssinxp  34228  rnxrnidres  34299  cossid  34370  fourierdlem42  40684
  Copyright terms: Public domain W3C validator