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Theorem triin 5282
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 5001 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3483 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 r19.26 3109 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) ↔ (∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵))
4 trss 5276 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
54imp 406 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
65ralimi 3081 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
73, 6sylbir 235 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
8 ssiin 5060 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
97, 8sylibr 234 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
102, 9sylan2b 594 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1110ralrimiva 3144 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
12 dftr3 5271 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1311, 12sylibr 234 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wcel 2106  wral 3059  Vcvv 3478  wss 3963   ciin 4997  Tr wtr 5265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-v 3480  df-ss 3980  df-uni 4913  df-iin 4999  df-tr 5266
This theorem is referenced by:  trint  5283
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