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Theorem bj-restsnid 36458
Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 36453 and bj-restsnss 36454. (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 3996 . . 3 𝐴𝐴
2 bj-restsnss 36454 . . 3 ((𝐴 ∈ V ∧ 𝐴𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
31, 2mpan2 688 . 2 (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4 df-rest 17367 . . . . 5 t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧𝑥 ↦ (𝑧𝑦)))
54reldmmpo 7535 . . . 4 Rel dom ↾t
65ovprc2 7441 . . 3 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7 snprc 4713 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
87biimpi 215 . . 3 𝐴 ∈ V → {𝐴} = ∅)
96, 8eqtr4d 2767 . 2 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
103, 9pm2.61i 182 1 ({𝐴} ↾t 𝐴) = {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3466  cin 3939  wss 3940  c0 4314  {csn 4620  cmpt 5221  ran crn 5667  (class class class)co 7401  t crest 17365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-rest 17367
This theorem is referenced by: (None)
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