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 32698
 Description: The elementwise intersection on the singleton on a class by that class is the singleton on that class. Special case of bj-restsn 32693 and bj-restsnss 32694. (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 3605 . . 3 𝐴𝐴
2 bj-restsnss 32694 . . 3 ((𝐴 ∈ V ∧ 𝐴𝐴) → ({𝐴} ↾t 𝐴) = {𝐴})
31, 2mpan2 706 . 2 (𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
4 df-rest 16007 . . . . 5 t = (𝑥 ∈ V, 𝑦 ∈ V ↦ ran (𝑧𝑥 ↦ (𝑧𝑦)))
54reldmmpt2 6727 . . . 4 Rel dom ↾t
65ovprc2 6641 . . 3 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = ∅)
7 snprc 4225 . . . 4 𝐴 ∈ V ↔ {𝐴} = ∅)
87biimpi 206 . . 3 𝐴 ∈ V → {𝐴} = ∅)
96, 8eqtr4d 2658 . 2 𝐴 ∈ V → ({𝐴} ↾t 𝐴) = {𝐴})
103, 9pm2.61i 176 1 ({𝐴} ↾t 𝐴) = {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1480   ∈ wcel 1987  Vcvv 3186   ∩ cin 3555   ⊆ wss 3556  ∅c0 3893  {csn 4150   ↦ cmpt 4675  ran crn 5077  (class class class)co 6607   ↾t crest 16005 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-rest 16007 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator