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Theorem bj-restsn 36470
Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 36473 and bj-restsnid 36475. (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 5424 . . . 4 {𝑌} ∈ V
2 elrest 17382 . . . 4 (({𝑌} ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
31, 2mpan 687 . . 3 (𝐴𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
4 velsn 4639 . . . . 5 (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 = (𝑦𝐴))
5 ineq1 4200 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝐴) = (𝑌𝐴))
65sneqd 4635 . . . . . 6 (𝑦 = 𝑌 → {(𝑦𝐴)} = {(𝑌𝐴)})
76eleq2d 2813 . . . . 5 (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 ∈ {(𝑌𝐴)}))
84, 7bitr3id 285 . . . 4 (𝑦 = 𝑌 → (𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
98rexsng 4673 . . 3 (𝑌𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
103, 9sylan9bbr 510 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
1110eqrdv 2724 1 ((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468  cin 3942  {csn 4623  (class class class)co 7405  t crest 17375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-rest 17377
This theorem is referenced by:  bj-restsnss  36471  bj-restsnss2  36472
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