Theorem triin 5202

Description: An indexed intersection of a class of transitive sets is transitive. (Contributed by BJ, 3-Oct-2022.)

Assertion

Ref	Expression
triin	⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵)

Proof of Theorem triin

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliin 4926	. . . . 5 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
2	1	elv 3428	. . . 4 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵)
3		r19.26 3094	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ (∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵))
4		trss 5196	. . . . . . . 8 ⊢ (Tr 𝐵 → (𝑦 ∈ 𝐵 → 𝑦 ⊆ 𝐵))
5	4	imp 406	. . . . . . 7 ⊢ ((Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ⊆ 𝐵)
6	5	ralimi 3086	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (Tr 𝐵 ∧ 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵)
7	3, 6	sylbir 234	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵)
8		ssiin 4981	. . . . 5 ⊢ (𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑦 ⊆ 𝐵)
9	7, 8	sylibr 233	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
10	2, 9	sylan2b 593	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 Tr 𝐵 ∧ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵) → 𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
11	10	ralrimiva 3107	. 2 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
12		dftr3 5191	. 2 ⊢ (Tr ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐵𝑦 ⊆ ∩ 𝑥 ∈ 𝐴 𝐵)
13	11, 12	sylibr 233	1 ⊢ (∀𝑥 ∈ 𝐴 Tr 𝐵 → Tr ∩ 𝑥 ∈ 𝐴 𝐵)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ⊆ wss 3883  ∩ ciin 4922  Tr wtr 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-uni 4837  df-iin 4924  df-tr 5188

This theorem is referenced by:  trint  5203