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Theorem triin 5163
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 4900 . . . . 5 (𝑦 ∈ V → (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵))
21elv 3478 . . . 4 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
3 r19.26 3157 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) ↔ (∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵))
4 trss 5157 . . . . . . . 8 (Tr 𝐵 → (𝑦𝐵𝑦𝐵))
54imp 409 . . . . . . 7 ((Tr 𝐵𝑦𝐵) → 𝑦𝐵)
65ralimi 3147 . . . . . 6 (∀𝑥𝐴 (Tr 𝐵𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
73, 6sylbir 237 . . . . 5 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → ∀𝑥𝐴 𝑦𝐵)
8 ssiin 4955 . . . . 5 (𝑦 𝑥𝐴 𝐵 ↔ ∀𝑥𝐴 𝑦𝐵)
97, 8sylibr 236 . . . 4 ((∀𝑥𝐴 Tr 𝐵 ∧ ∀𝑥𝐴 𝑦𝐵) → 𝑦 𝑥𝐴 𝐵)
102, 9sylan2b 595 . . 3 ((∀𝑥𝐴 Tr 𝐵𝑦 𝑥𝐴 𝐵) → 𝑦 𝑥𝐴 𝐵)
1110ralrimiva 3169 . 2 (∀𝑥𝐴 Tr 𝐵 → ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
12 dftr3 5152 . 2 (Tr 𝑥𝐴 𝐵 ↔ ∀𝑦 𝑥𝐴 𝐵𝑦 𝑥𝐴 𝐵)
1311, 12sylibr 236 1 (∀𝑥𝐴 Tr 𝐵 → Tr 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398  wcel 2114  wral 3125  Vcvv 3473  wss 3913   ciin 4896  Tr wtr 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-v 3475  df-in 3920  df-ss 3930  df-uni 4815  df-iin 4898  df-tr 5149
This theorem is referenced by:  trint  5164
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