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Theorem bj-restsnid 35906
Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 35901 and bj-restsnss 35902. (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 4003 . . 3 𝐴𝐴
2 bj-restsnss 35902 . . 3 ((𝐴 ∈ V ∧ 𝐴𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
31, 2mpan2 690 . 2 (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4 df-rest 17364 . . . . 5 t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧𝑥 ↦ (𝑧𝑦)))
54reldmmpo 7538 . . . 4 Rel dom ↾t
65ovprc2 7444 . . 3 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7 snprc 4720 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
87biimpi 215 . . 3 𝐴 ∈ V → {𝐴} = ∅)
96, 8eqtr4d 2776 . 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 1542  wcel 2107  Vcvv 3475  cin 3946  wss 3947  c0 4321  {csn 4627  cmpt 5230  ran crn 5676  (class class class)co 7404  t crest 17362
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-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  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-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-rest 17364
This theorem is referenced by: (None)
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