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Theorem cotrintab 37399
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 → (𝑥𝑥) ⊆ 𝑥)
Assertion
Ref Expression
cotrintab ( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑}

Proof of Theorem cotrintab
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cotr 5467 . 2 (( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑} ↔ ∀𝑢𝑤𝑣((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣))
2 pm3.43 905 . . . . . 6 (((𝜑𝑢𝑥𝑤) ∧ (𝜑𝑤𝑥𝑣)) → (𝜑 → (𝑢𝑥𝑤𝑤𝑥𝑣)))
3 cotrintab.min . . . . . . 7 (𝜑 → (𝑥𝑥) ⊆ 𝑥)
4 cotr 5467 . . . . . . . 8 ((𝑥𝑥) ⊆ 𝑥 ↔ ∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
54biimpi 206 . . . . . . 7 ((𝑥𝑥) ⊆ 𝑥 → ∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
6 2sp 2054 . . . . . . . 8 (∀𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
76sps 2053 . . . . . . 7 (∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
83, 5, 73syl 18 . . . . . 6 (𝜑 → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
92, 8sylcom 30 . . . . 5 (((𝜑𝑢𝑥𝑤) ∧ (𝜑𝑤𝑥𝑣)) → (𝜑𝑢𝑥𝑣))
109alanimi 1741 . . . 4 ((∀𝑥(𝜑𝑢𝑥𝑤) ∧ ∀𝑥(𝜑𝑤𝑥𝑣)) → ∀𝑥(𝜑𝑢𝑥𝑣))
11 opex 4893 . . . . . . 7 𝑢, 𝑤⟩ ∈ V
1211elintab 4452 . . . . . 6 (⟨𝑢, 𝑤⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
13 df-br 4614 . . . . . 6 (𝑢 {𝑥𝜑}𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ {𝑥𝜑})
14 df-br 4614 . . . . . . . 8 (𝑢𝑥𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ 𝑥)
1514imbi2i 326 . . . . . . 7 ((𝜑𝑢𝑥𝑤) ↔ (𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
1615albii 1744 . . . . . 6 (∀𝑥(𝜑𝑢𝑥𝑤) ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
1712, 13, 163bitr4i 292 . . . . 5 (𝑢 {𝑥𝜑}𝑤 ↔ ∀𝑥(𝜑𝑢𝑥𝑤))
18 opex 4893 . . . . . . 7 𝑤, 𝑣⟩ ∈ V
1918elintab 4452 . . . . . 6 (⟨𝑤, 𝑣⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
20 df-br 4614 . . . . . 6 (𝑤 {𝑥𝜑}𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ {𝑥𝜑})
21 df-br 4614 . . . . . . . 8 (𝑤𝑥𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ 𝑥)
2221imbi2i 326 . . . . . . 7 ((𝜑𝑤𝑥𝑣) ↔ (𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
2322albii 1744 . . . . . 6 (∀𝑥(𝜑𝑤𝑥𝑣) ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
2419, 20, 233bitr4i 292 . . . . 5 (𝑤 {𝑥𝜑}𝑣 ↔ ∀𝑥(𝜑𝑤𝑥𝑣))
2517, 24anbi12i 732 . . . 4 ((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) ↔ (∀𝑥(𝜑𝑢𝑥𝑤) ∧ ∀𝑥(𝜑𝑤𝑥𝑣)))
26 opex 4893 . . . . . 6 𝑢, 𝑣⟩ ∈ V
2726elintab 4452 . . . . 5 (⟨𝑢, 𝑣⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
28 df-br 4614 . . . . 5 (𝑢 {𝑥𝜑}𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ {𝑥𝜑})
29 df-br 4614 . . . . . . 7 (𝑢𝑥𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑥)
3029imbi2i 326 . . . . . 6 ((𝜑𝑢𝑥𝑣) ↔ (𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
3130albii 1744 . . . . 5 (∀𝑥(𝜑𝑢𝑥𝑣) ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
3227, 28, 313bitr4i 292 . . . 4 (𝑢 {𝑥𝜑}𝑣 ↔ ∀𝑥(𝜑𝑢𝑥𝑣))
3310, 25, 323imtr4i 281 . . 3 ((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣)
3433gen2 1720 . 2 𝑤𝑣((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣)
351, 34mpgbir 1723 1 ( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1478  wcel 1987  {cab 2607  wss 3555  cop 4154   cint 4440   class class class wbr 4613  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-int 4441  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-co 5083
This theorem is referenced by:  dfrtrcl5  37414
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