Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-restsn Structured version   Visualization version   GIF version

Theorem bj-restsn 33367
Description: An elementwise intersection on the singleton on a set is the singleton on the intersection by that set. Generalization of bj-restsn0 33370 and bj-restsnid 33372. (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 5036 . . . 4 {𝑌} ∈ V
2 elrest 16296 . . . 4 (({𝑌} ∈ V ∧ 𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
31, 2mpan 670 . . 3 (𝐴𝑊 → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ ∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴)))
4 velsn 4332 . . . . 5 (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 = (𝑦𝐴))
5 ineq1 3958 . . . . . . 7 (𝑦 = 𝑌 → (𝑦𝐴) = (𝑌𝐴))
65sneqd 4328 . . . . . 6 (𝑦 = 𝑌 → {(𝑦𝐴)} = {(𝑌𝐴)})
76eleq2d 2836 . . . . 5 (𝑦 = 𝑌 → (𝑥 ∈ {(𝑦𝐴)} ↔ 𝑥 ∈ {(𝑌𝐴)}))
84, 7syl5bbr 274 . . . 4 (𝑦 = 𝑌 → (𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
98rexsng 4357 . . 3 (𝑌𝑉 → (∃𝑦 ∈ {𝑌}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
103, 9sylan9bbr 500 . 2 ((𝑌𝑉𝐴𝑊) → (𝑥 ∈ ({𝑌} ↾t 𝐴) ↔ 𝑥 ∈ {(𝑌𝐴)}))
1110eqrdv 2769 1 ((𝑌𝑉𝐴𝑊) → ({𝑌} ↾t 𝐴) = {(𝑌𝐴)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  wrex 3062  Vcvv 3351  cin 3722  {csn 4316  (class class class)co 6793  t crest 16289
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-rest 16291
This theorem is referenced by:  bj-restsnss  33368  bj-restsnss2  33369
  Copyright terms: Public domain W3C validator