MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  triin Structured version   Visualization version   GIF version

Theorem triin 5243
Description: An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)
Assertion
Ref Expression
triin (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)

Proof of Theorem triin
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliin 4963 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3453 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 r19.26 3111 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) ↔ (∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵))
4 trss 5237 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
54imp 408 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
65ralimi 3083 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
73, 6sylbir 234 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
8 ssiin 5019 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
97, 8sylibr 233 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
102, 9sylan2b 595 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1110ralrimiva 3140 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
12 dftr3 5232 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1311, 12sylibr 233 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wcel 2107  wral 3061  Vcvv 3447  wss 3914   ciin 4959  Tr wtr 5226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-v 3449  df-in 3921  df-ss 3931  df-uni 4870  df-iin 4961  df-tr 5227
This theorem is referenced by:  trint  5244
  Copyright terms: Public domain W3C validator