Theorem cotrintab 39480

Description: The intersection of a class is a transitive relation if membership in the class implies the member is a transitive relation. (Contributed by RP, 28-Oct-2020.)

Hypothesis

Ref	Expression
cotrintab.min	⊢ (𝜑 → (𝑥 ∘ 𝑥) ⊆ 𝑥)

Assertion

Ref	Expression
cotrintab	⊢ (∩ {𝑥 ∣ 𝜑} ∘ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑥 ∣ 𝜑}

Proof of Theorem cotrintab

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cotr 5855	. 2 ⊢ ((∩ {𝑥 ∣ 𝜑} ∘ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑢∀𝑤∀𝑣((𝑢∩ {𝑥 ∣ 𝜑}𝑤 ∧ 𝑤∩ {𝑥 ∣ 𝜑}𝑣) → 𝑢∩ {𝑥 ∣ 𝜑}𝑣))
2		pm3.43 474	. . . . . 6 ⊢ (((𝜑 → 𝑢𝑥𝑤) ∧ (𝜑 → 𝑤𝑥𝑣)) → (𝜑 → (𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣)))
3		cotrintab.min	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∘ 𝑥) ⊆ 𝑥)
4		cotr 5855	. . . . . . . 8 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 ↔ ∀𝑢∀𝑤∀𝑣((𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣) → 𝑢𝑥𝑣))
5	4	biimpi 217	. . . . . . 7 ⊢ ((𝑥 ∘ 𝑥) ⊆ 𝑥 → ∀𝑢∀𝑤∀𝑣((𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣) → 𝑢𝑥𝑣))
6		2sp 2151	. . . . . . . 8 ⊢ (∀𝑤∀𝑣((𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣) → 𝑢𝑥𝑣))
7	6	sps 2150	. . . . . . 7 ⊢ (∀𝑢∀𝑤∀𝑣((𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣) → 𝑢𝑥𝑣))
8	3, 5, 7	3syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑢𝑥𝑤 ∧ 𝑤𝑥𝑣) → 𝑢𝑥𝑣))
9	2, 8	sylcom 30	. . . . 5 ⊢ (((𝜑 → 𝑢𝑥𝑤) ∧ (𝜑 → 𝑤𝑥𝑣)) → (𝜑 → 𝑢𝑥𝑣))
10	9	alanimi 1802	. . . 4 ⊢ ((∀𝑥(𝜑 → 𝑢𝑥𝑤) ∧ ∀𝑥(𝜑 → 𝑤𝑥𝑣)) → ∀𝑥(𝜑 → 𝑢𝑥𝑣))
11		opex 5255	. . . . . . 7 ⊢ ⟨𝑢, 𝑤⟩ ∈ V
12	11	elintab 4799	. . . . . 6 ⊢ (⟨𝑢, 𝑤⟩ ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
13		df-br 4969	. . . . . 6 ⊢ (𝑢∩ {𝑥 ∣ 𝜑}𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ ∩ {𝑥 ∣ 𝜑})
14		df-br 4969	. . . . . . . 8 ⊢ (𝑢𝑥𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ 𝑥)
15	14	imbi2i 337	. . . . . . 7 ⊢ ((𝜑 → 𝑢𝑥𝑤) ↔ (𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
16	15	albii 1805	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑢𝑥𝑤) ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
17	12, 13, 16	3bitr4i 304	. . . . 5 ⊢ (𝑢∩ {𝑥 ∣ 𝜑}𝑤 ↔ ∀𝑥(𝜑 → 𝑢𝑥𝑤))
18		opex 5255	. . . . . . 7 ⊢ ⟨𝑤, 𝑣⟩ ∈ V
19	18	elintab 4799	. . . . . 6 ⊢ (⟨𝑤, 𝑣⟩ ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
20		df-br 4969	. . . . . 6 ⊢ (𝑤∩ {𝑥 ∣ 𝜑}𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ ∩ {𝑥 ∣ 𝜑})
21		df-br 4969	. . . . . . . 8 ⊢ (𝑤𝑥𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ 𝑥)
22	21	imbi2i 337	. . . . . . 7 ⊢ ((𝜑 → 𝑤𝑥𝑣) ↔ (𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
23	22	albii 1805	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝑤𝑥𝑣) ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
24	19, 20, 23	3bitr4i 304	. . . . 5 ⊢ (𝑤∩ {𝑥 ∣ 𝜑}𝑣 ↔ ∀𝑥(𝜑 → 𝑤𝑥𝑣))
25	17, 24	anbi12i 626	. . . 4 ⊢ ((𝑢∩ {𝑥 ∣ 𝜑}𝑤 ∧ 𝑤∩ {𝑥 ∣ 𝜑}𝑣) ↔ (∀𝑥(𝜑 → 𝑢𝑥𝑤) ∧ ∀𝑥(𝜑 → 𝑤𝑥𝑣)))
26		opex 5255	. . . . . 6 ⊢ ⟨𝑢, 𝑣⟩ ∈ V
27	26	elintab 4799	. . . . 5 ⊢ (⟨𝑢, 𝑣⟩ ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
28		df-br 4969	. . . . 5 ⊢ (𝑢∩ {𝑥 ∣ 𝜑}𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ ∩ {𝑥 ∣ 𝜑})
29		df-br 4969	. . . . . . 7 ⊢ (𝑢𝑥𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑥)
30	29	imbi2i 337	. . . . . 6 ⊢ ((𝜑 → 𝑢𝑥𝑣) ↔ (𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
31	30	albii 1805	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝑢𝑥𝑣) ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
32	27, 28, 31	3bitr4i 304	. . . 4 ⊢ (𝑢∩ {𝑥 ∣ 𝜑}𝑣 ↔ ∀𝑥(𝜑 → 𝑢𝑥𝑣))
33	10, 25, 32	3imtr4i 293	. . 3 ⊢ ((𝑢∩ {𝑥 ∣ 𝜑}𝑤 ∧ 𝑤∩ {𝑥 ∣ 𝜑}𝑣) → 𝑢∩ {𝑥 ∣ 𝜑}𝑣)
34	33	gen2 1782	. 2 ⊢ ∀𝑤∀𝑣((𝑢∩ {𝑥 ∣ 𝜑}𝑤 ∧ 𝑤∩ {𝑥 ∣ 𝜑}𝑣) → 𝑢∩ {𝑥 ∣ 𝜑}𝑣)
35	1, 34	mpgbir 1785	1 ⊢ (∩ {𝑥 ∣ 𝜑} ∘ ∩ {𝑥 ∣ 𝜑}) ⊆ ∩ {𝑥 ∣ 𝜑}

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1523   ∈ wcel 2083  {cab 2777   ⊆ wss 3865  ⟨cop 4484  ∩ cint 4788   class class class wbr 4968   ∘ ccom 5454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-int 4789  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-co 5459

This theorem is referenced by:  dfrtrcl5  39495