Theorem trficl 43693

Description: The class of all transitive relations has the finite intersection property. (Contributed by RP, 1-Jan-2020.) (Proof shortened by RP, 3-Jan-2020.)

Hypothesis

Ref	Expression
trficl.a	⊢ 𝐴 = {𝑧 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧}

Assertion

Ref	Expression
trficl	⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝐴

Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem trficl

Step	Hyp	Ref	Expression
1		trficl.a	. 2 ⊢ 𝐴 = {𝑧 ∣ (𝑧 ∘ 𝑧) ⊆ 𝑧}
2		vex 3463	. . 3 ⊢ 𝑥 ∈ V
3	2	inex1 5287	. 2 ⊢ (𝑥 ∩ 𝑦) ∈ V
4		id 22	. . . 4 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → 𝑧 = (𝑥 ∩ 𝑦))
5	4, 4	coeq12d 5844	. . 3 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → (𝑧 ∘ 𝑧) = ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)))
6	5, 4	sseq12d 3992	. 2 ⊢ (𝑧 = (𝑥 ∩ 𝑦) → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦)))
7		id 22	. . . 4 ⊢ (𝑧 = 𝑥 → 𝑧 = 𝑥)
8	7, 7	coeq12d 5844	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∘ 𝑧) = (𝑥 ∘ 𝑥))
9	8, 7	sseq12d 3992	. 2 ⊢ (𝑧 = 𝑥 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑥 ∘ 𝑥) ⊆ 𝑥))
10		id 22	. . . 4 ⊢ (𝑧 = 𝑦 → 𝑧 = 𝑦)
11	10, 10	coeq12d 5844	. . 3 ⊢ (𝑧 = 𝑦 → (𝑧 ∘ 𝑧) = (𝑦 ∘ 𝑦))
12	11, 10	sseq12d 3992	. 2 ⊢ (𝑧 = 𝑦 → ((𝑧 ∘ 𝑧) ⊆ 𝑧 ↔ (𝑦 ∘ 𝑦) ⊆ 𝑦))
13		trin2 6112	. 2 ⊢ (((𝑥 ∘ 𝑥) ⊆ 𝑥 ∧ (𝑦 ∘ 𝑦) ⊆ 𝑦) → ((𝑥 ∩ 𝑦) ∘ (𝑥 ∩ 𝑦)) ⊆ (𝑥 ∩ 𝑦))
14	1, 3, 6, 9, 12, 13	cllem0 43590	1 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∩ 𝑦) ∈ 𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  Vcvv 3459   ∩ cin 3925   ⊆ wss 3926   ∘ ccom 5658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-co 5663

This theorem is referenced by: (None)