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Theorem bj-restn0 33018
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 3923 . . . 4 (𝑋 ≠ ∅ ↔ ∃𝑦 𝑦𝑋)
2 vex 3198 . . . . . . . . . . 11 𝑦 ∈ V
32inex1 4790 . . . . . . . . . 10 (𝑦𝐴) ∈ V
43isseti 3204 . . . . . . . . 9 𝑥 𝑥 = (𝑦𝐴)
54jctr 564 . . . . . . . 8 (𝑦𝑋 → (𝑦𝑋 ∧ ∃𝑥 𝑥 = (𝑦𝐴)))
65eximi 1760 . . . . . . 7 (∃𝑦 𝑦𝑋 → ∃𝑦(𝑦𝑋 ∧ ∃𝑥 𝑥 = (𝑦𝐴)))
7 df-rex 2915 . . . . . . 7 (∃𝑦𝑋𝑥 𝑥 = (𝑦𝐴) ↔ ∃𝑦(𝑦𝑋 ∧ ∃𝑥 𝑥 = (𝑦𝐴)))
86, 7sylibr 224 . . . . . 6 (∃𝑦 𝑦𝑋 → ∃𝑦𝑋𝑥 𝑥 = (𝑦𝐴))
9 rexcom4 3220 . . . . . 6 (∃𝑦𝑋𝑥 𝑥 = (𝑦𝐴) ↔ ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴))
108, 9sylib 208 . . . . 5 (∃𝑦 𝑦𝑋 → ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴))
1110a1i 11 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦 𝑦𝑋 → ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴)))
121, 11syl5bi 232 . . 3 ((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → ∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴)))
13 elrest 16069 . . . . 5 ((𝑋𝑉𝐴𝑊) → (𝑥 ∈ (𝑋t 𝐴) ↔ ∃𝑦𝑋 𝑥 = (𝑦𝐴)))
1413biimprd 238 . . . 4 ((𝑋𝑉𝐴𝑊) → (∃𝑦𝑋 𝑥 = (𝑦𝐴) → 𝑥 ∈ (𝑋t 𝐴)))
1514eximdv 1844 . . 3 ((𝑋𝑉𝐴𝑊) → (∃𝑥𝑦𝑋 𝑥 = (𝑦𝐴) → ∃𝑥 𝑥 ∈ (𝑋t 𝐴)))
1612, 15syld 47 . 2 ((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → ∃𝑥 𝑥 ∈ (𝑋t 𝐴)))
17 n0 3923 . 2 ((𝑋t 𝐴) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑋t 𝐴))
1816, 17syl6ibr 242 1 ((𝑋𝑉𝐴𝑊) → (𝑋 ≠ ∅ → (𝑋t 𝐴) ≠ ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wex 1702  wcel 1988  wne 2791  wrex 2910  cin 3566  c0 3907  (class class class)co 6635  t crest 16062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-rest 16064
This theorem is referenced by:  bj-restn0b  33019
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