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

Theorem ideqg 5183
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 5159 . . . 4 Rel I
32brrelexi 5072 . . 3 (𝐴 I 𝐵𝐴 ∈ V)
41, 3anim12ci 588 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
5 eleq1 2675 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
65biimparc 502 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
76elexd 3186 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
8 simpl 471 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
97, 8jca 552 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 eqeq1 2613 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqeq2 2620 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
12 df-id 4943 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1310, 11, 12brabg 4909 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
144, 9, 13pm5.21nd 938 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  Vcvv 3172   class class class wbr 4577   I cid 4938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035
This theorem is referenced by:  ideq  5184  ididg  5185  restidsingOLD  5365  poleloe  5433  isof1oidb  6452  pltval  16729  tglngne  25163  tgelrnln  25243  opeldifid  28600  fourierdlem42  38846
  Copyright terms: Public domain W3C validator