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Theorem bj-restsn 33341
Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 33344 and bj-restsnid 33346. (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 5057 . . . 4 {𝑌} ∈ V
2 elrest 16290 . . . 4 (({𝑌} ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
31, 2mpan 708 . . 3 (𝐴𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
4 velsn 4337 . . . . 5 (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 = (𝑦𝐴))
5 ineq1 3950 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝐴) = (𝑌𝐴))
65sneqd 4333 . . . . . 6 (𝑦 = 𝑌 → {(𝑦𝐴)} = {(𝑌𝐴)})
76eleq2d 2825 . . . . 5 (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 ∈ {(𝑌𝐴)}))
84, 7syl5bbr 274 . . . 4 (𝑦 = 𝑌 → (𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
98rexsng 4363 . . 3 (𝑌𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
103, 9sylan9bbr 739 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
1110eqrdv 2758 1 ((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wrex 3051  Vcvv 3340  cin 3714  {csn 4321  (class class class)co 6813  t crest 16283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-rest 16285
This theorem is referenced by:  bj-restsnss  33342  bj-restsnss2  33343
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