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Theorem bj-restsn 32012
Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 32015 and bj-restsnid 32017. (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 4830 . . . 4 {𝑌} ∈ V
2 elrest 15857 . . . 4 (({𝑌} ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
31, 2mpan 701 . . 3 (𝐴𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
4 velsn 4140 . . . . 5 (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 = (𝑦𝐴))
5 ineq1 3768 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝐴) = (𝑌𝐴))
65sneqd 4136 . . . . . 6 (𝑦 = 𝑌 → {(𝑦𝐴)} = {(𝑌𝐴)})
76eleq2d 2672 . . . . 5 (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 ∈ {(𝑌𝐴)}))
84, 7syl5bbr 272 . . . 4 (𝑦 = 𝑌 → (𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
98rexsng 4165 . . 3 (𝑌𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
103, 9sylan9bbr 732 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
1110eqrdv 2607 1 ((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wrex 2896  Vcvv 3172  cin 3538  {csn 4124  (class class class)co 6527  t crest 15850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-rest 15852
This theorem is referenced by:  bj-restsnss  32013  bj-restsnss2  32014
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