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

Theorem ideqg 5429
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 5405 . . . 4 Rel I
32brrelexi 5315 . . 3 (𝐴 I 𝐵𝐴 ∈ V)
41, 3anim12ci 592 . 2 ((𝐵𝑉𝐴 I 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
5 eleq1 2827 . . . . 5 (𝐴 = 𝐵 → (𝐴𝑉𝐵𝑉))
65biimparc 505 . . . 4 ((𝐵𝑉𝐴 = 𝐵) → 𝐴𝑉)
76elexd 3354 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐴 ∈ V)
8 simpl 474 . . 3 ((𝐵𝑉𝐴 = 𝐵) → 𝐵𝑉)
97, 8jca 555 . 2 ((𝐵𝑉𝐴 = 𝐵) → (𝐴 ∈ V ∧ 𝐵𝑉))
10 eqeq1 2764 . . 3 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
11 eqeq2 2771 . . 3 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
12 df-id 5174 . . 3 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
1310, 11, 12brabg 5144 . 2 ((𝐴 ∈ V ∧ 𝐵𝑉) → (𝐴 I 𝐵𝐴 = 𝐵))
144, 9, 13pm5.21nd 979 1 (𝐵𝑉 → (𝐴 I 𝐵𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  Vcvv 3340   class class class wbr 4804   I cid 5173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273
This theorem is referenced by:  ideq  5430  ididg  5431  restidsingOLD  5617  poleloe  5685  isof1oidb  6737  pltval  17161  tglngne  25644  tgelrnln  25724  opeldifid  29719  ideq2  34402  idinxpss  34407  inxpssidinxp  34410  idinxpssinxp  34411  rnxrnidres  34482  cossid  34553  fourierdlem42  40869
  Copyright terms: Public domain W3C validator