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

Theorem elrid 6003
Description: Characterization of the elements of a restricted identity relation. (Contributed by BJ, 28-Aug-2022.) (Proof shortened by Peter Mazsa, 9-Sep-2022.)
Assertion
Ref Expression
elrid (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑋

Proof of Theorem elrid
StepHypRef Expression
1 df-res 5649 . . 3 ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
21eleq2i 2826 . 2 (𝐴 ∈ ( I ↾ 𝑋) ↔ 𝐴 ∈ ( I ∩ (𝑋 × V)))
3 elidinxp 6001 . 2 (𝐴 ∈ ( I ∩ (𝑋 × V)) ↔ ∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩)
4 inv1 4358 . . 3 (𝑋 ∩ V) = 𝑋
54rexeqi 3311 . 2 (∃𝑥 ∈ (𝑋 ∩ V)𝐴 = ⟨𝑥, 𝑥⟩ ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
62, 3, 53bitri 297 1 (𝐴 ∈ ( I ↾ 𝑋) ↔ ∃𝑥𝑋 𝐴 = ⟨𝑥, 𝑥⟩)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  wrex 3070  Vcvv 3447  cin 3913  cop 4596   I cid 5534   × cxp 5635  cres 5639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-res 5649
This theorem is referenced by:  idinxpres  6004  idrefALT  6069  elid  6155
  Copyright terms: Public domain W3C validator