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

Theorem eliind 39554
Description: Membership in indexed intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
eliind.a (𝜑𝐴 𝑥𝐵 𝐶)
eliind.k (𝜑𝐾𝐵)
eliind.d (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))
Assertion
Ref Expression
eliind (𝜑𝐴𝐷)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐷   𝑥,𝐾
Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)

Proof of Theorem eliind
StepHypRef Expression
1 eliind.k . 2 (𝜑𝐾𝐵)
2 eliind.a . . 3 (𝜑𝐴 𝑥𝐵 𝐶)
3 eliin 4557 . . . 4 (𝐴 𝑥𝐵 𝐶 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
42, 3syl 17 . . 3 (𝜑 → (𝐴 𝑥𝐵 𝐶 ↔ ∀𝑥𝐵 𝐴𝐶))
52, 4mpbid 222 . 2 (𝜑 → ∀𝑥𝐵 𝐴𝐶)
6 eliind.d . . 3 (𝑥 = 𝐾 → (𝐴𝐶𝐴𝐷))
76rspcva 3338 . 2 ((𝐾𝐵 ∧ ∀𝑥𝐵 𝐴𝐶) → 𝐴𝐷)
81, 5, 7syl2anc 694 1 (𝜑𝐴𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1523  wcel 2030  wral 2941   ciin 4553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-iin 4555
This theorem is referenced by:  iooiinioc  40101  hspdifhsp  41151  smflimlem3  41302  smfsuplem1  41338  smflimsuplem4  41350
  Copyright terms: Public domain W3C validator