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Theorem bj-restn0 34275
Description: An elementwise intersection on a nonempty family is nonempty. (Contributed by BJ, 27-Apr-2021.)
Assertion
Ref Expression
bj-restn0 ((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))

Proof of Theorem bj-restn0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4307 . . . 4 (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦𝑋)
2 vex 3495 . . . . . . . . . . 11 𝑦 ∈ V
32inex1 5212 . . . . . . . . . 10 (𝑦𝐴) ∈ V
43isseti 3506 . . . . . . . . 9 𝑥 𝑥 = (𝑦𝐴)
54jctr 525 . . . . . . . 8 (𝑦𝑋 → (𝑦𝑋 ∧ ∃𝑥 𝑥 = (𝑦𝐴)))
65eximi 1826 . . . . . . 7 (∃𝑦 𝑦𝑋 → ∃𝑦(𝑦𝑋 ∧ ∃𝑥 𝑥 = (𝑦𝐴)))
7 df-rex 3141 . . . . . . 7 (∃𝑦𝑋𝑥 𝑥 = (𝑦𝐴) ↔ ∃𝑦(𝑦𝑋 ∧ ∃𝑥 𝑥 = (𝑦𝐴)))
86, 7sylibr 235 . . . . . 6 (∃𝑦 𝑦𝑋 → ∃𝑦𝑋𝑥 𝑥 = (𝑦𝐴))
9 rexcom4 3246 . . . . . 6 (∃𝑦𝑋𝑥 𝑥 = (𝑦𝐴) ↔ ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴))
108, 9sylib 219 . . . . 5 (∃𝑦 𝑦𝑋 → ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴))
1110a1i 11 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦 𝑦𝑋 → ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴)))
121, 11syl5bi 243 . . 3 ((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴)))
13 elrest 16689 . . . . 5 ((𝑋𝑉𝐴𝑊) → (𝑥 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝑥 = (𝑦𝐴)))
1413biimprd 249 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦𝑋 𝑥 = (𝑦𝐴) → 𝑥 ∈ (𝑋t 𝐴)))
1514eximdv 1909 . . 3 ((𝑋𝑉𝐴𝑊) → (∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴) → ∃𝑥 𝑥 ∈ (𝑋t 𝐴)))
1612, 15syld 47 . 2 ((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → ∃𝑥 𝑥 ∈ (𝑋t 𝐴)))
17 n0 4307 . 2 ((𝑋t 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑋t 𝐴))
1816, 17syl6ibr 253 1 ((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wne 3013  wrex 3136  cin 3932  c0 4288  (class class class)co 7145  t crest 16682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-rest 16684
This theorem is referenced by:  bj-restn0b  34276
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