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

Theorem elimasng 6086
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.) TODO: replace existing usages by usages of elimasng1 6084, remove, and relabel elimasng1 6084 to "elimasng".
Assertion
Ref Expression
elimasng ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))

Proof of Theorem elimasng
StepHypRef Expression
1 elimasng1 6084 . 2 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ 𝐵𝐴𝐶))
2 df-br 5148 . 2 (𝐵𝐴𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴)
31, 2bitrdi 286 1 ((𝐵𝑉𝐶𝑊) → (𝐶 ∈ (𝐴 “ {𝐵}) ↔ ⟨𝐵, 𝐶⟩ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wcel 2104  {csn 4627  cop 4633   class class class wbr 5147  cima 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688
This theorem is referenced by:  elimasn  6087  inimasn  6154  dffv3  6886  fvimacnv  7053  fvrnressn  7160  elecg  8748  imasnopn  23414  imasncld  23415  imasncls  23416  ustelimasn  23947  blval2  24291  elbl4  24292  scutval  27538  iunsnima2  32115  1stpreimas  32194  opelco3  35050  funpartfv  35221  eltail  35562  elecALTV  37437  brtrclfv2  42780  frege77d  42799  dfafv23  46259
  Copyright terms: Public domain W3C validator