Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-restsnid Structured version   Visualization version   GIF version

Theorem bj-restsnid 34502
Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 34497 and bj-restsnss 34498. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restsnid ({𝐴} ↾t 𝐴) = {𝐴}

Proof of Theorem bj-restsnid
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3937 . . 3 𝐴𝐴
2 bj-restsnss 34498 . . 3 ((𝐴 ∈ V ∧ 𝐴𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
31, 2mpan2 690 . 2 (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4 df-rest 16688 . . . . 5 t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧𝑥 ↦ (𝑧𝑦)))
54reldmmpo 7264 . . . 4 Rel dom ↾t
65ovprc2 7175 . . 3 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7 snprc 4613 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
87biimpi 219 . . 3 𝐴 ∈ V → {𝐴} = ∅)
96, 8eqtr4d 2836 . 2 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
103, 9pm2.61i 185 1 ({𝐴} ↾t 𝐴) = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1538  wcel 2111  Vcvv 3441  cin 3880  wss 3881  c0 4243  {csn 4525  cmpt 5110  ran crn 5520  (class class class)co 7135  t crest 16686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-rest 16688
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator