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Theorem triin 5283
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 5003 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3481 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 r19.26 3112 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) ↔ (∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵))
4 trss 5277 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
54imp 408 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
65ralimi 3084 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
73, 6sylbir 234 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
8 ssiin 5059 . . . . 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 3147 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
12 dftr3 5272 . 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 3062  Vcvv 3475  wss 3949   ciin 4999  Tr wtr 5266
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 3063  df-v 3477  df-in 3956  df-ss 3966  df-uni 4910  df-iin 5001  df-tr 5267
This theorem is referenced by:  trint  5284
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