Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  eliinid Structured version   Visualization version   GIF version

Theorem eliinid 41669
 Description: Membership in an indexed intersection implies membership in any intersected set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Assertion
Ref Expression
eliinid ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem eliinid
StepHypRef Expression
1 simpl 486 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴 𝑥𝐵 𝐶)
2 eliin 4910 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
32adantr 484 . . 3 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
41, 3mpbid 235 . 2 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → ∀𝑥𝐵 𝐴𝐶)
5 rspa 3201 . 2 ((∀𝑥𝐵 𝐴𝐶𝑥𝐵) → 𝐴𝐶)
64, 5sylancom 591 1 ((𝐴 𝑥𝐵 𝐶𝑥𝐵) → 𝐴𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2115  ∀wral 3133  ∩ ciin 4906 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-iin 4908 This theorem is referenced by:  iinssiin  41686  fnlimfvre  42242  smflimlem2  43331  smflimmpt  43367  smfsuplem1  43368  smfsupmpt  43372  smfsupxr  43373  smfinflem  43374  smfinfmpt  43376  smflimsuplem4  43380  smflimsupmpt  43386  smfliminfmpt  43389
 Copyright terms: Public domain W3C validator