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Theorem bj-restsn 35951
Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 35954 and bj-restsnid 35956. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restsn ((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})

Proof of Theorem bj-restsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snex 5430 . . . 4 {𝑌} ∈ V
2 elrest 17369 . . . 4 (({𝑌} ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
31, 2mpan 688 . . 3 (𝐴𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
4 velsn 4643 . . . . 5 (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 = (𝑦𝐴))
5 ineq1 4204 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝐴) = (𝑌𝐴))
65sneqd 4639 . . . . . 6 (𝑦 = 𝑌 → {(𝑦𝐴)} = {(𝑌𝐴)})
76eleq2d 2819 . . . . 5 (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 ∈ {(𝑌𝐴)}))
84, 7bitr3id 284 . . . 4 (𝑦 = 𝑌 → (𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
98rexsng 4677 . . 3 (𝑌𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
103, 9sylan9bbr 511 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
1110eqrdv 2730 1 ((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  Vcvv 3474  cin 3946  {csn 4627  (class class class)co 7405  t crest 17362
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7408  df-oprab 7409  df-mpo 7410  df-rest 17364
This theorem is referenced by:  bj-restsnss  35952  bj-restsnss2  35953
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