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Theorem cotrintab 40312
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 5939 . 2 (( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑} ↔ ∀𝑢𝑤𝑣((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣))
2 pm3.43 477 . . . . . 6 (((𝜑𝑢𝑥𝑤) ∧ (𝜑𝑤𝑥𝑣)) → (𝜑 → (𝑢𝑥𝑤𝑤𝑥𝑣)))
3 cotrintab.min . . . . . . 7 (𝜑 → (𝑥𝑥) ⊆ 𝑥)
4 cotr 5939 . . . . . . . 8 ((𝑥𝑥) ⊆ 𝑥 ↔ ∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
54biimpi 219 . . . . . . 7 ((𝑥𝑥) ⊆ 𝑥 → ∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
6 2sp 2183 . . . . . . . 8 (∀𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
76sps 2182 . . . . . . 7 (∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
83, 5, 73syl 18 . . . . . 6 (𝜑 → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
92, 8sylcom 30 . . . . 5 (((𝜑𝑢𝑥𝑤) ∧ (𝜑𝑤𝑥𝑣)) → (𝜑𝑢𝑥𝑣))
109alanimi 1818 . . . 4 ((∀𝑥(𝜑𝑢𝑥𝑤) ∧ ∀𝑥(𝜑𝑤𝑥𝑣)) → ∀𝑥(𝜑𝑢𝑥𝑣))
11 opex 5321 . . . . . . 7 𝑢, 𝑤⟩ ∈ V
1211elintab 4849 . . . . . 6 (⟨𝑢, 𝑤⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
13 df-br 5031 . . . . . 6 (𝑢 {𝑥𝜑}𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ {𝑥𝜑})
14 df-br 5031 . . . . . . . 8 (𝑢𝑥𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ 𝑥)
1514imbi2i 339 . . . . . . 7 ((𝜑𝑢𝑥𝑤) ↔ (𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
1615albii 1821 . . . . . 6 (∀𝑥(𝜑𝑢𝑥𝑤) ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
1712, 13, 163bitr4i 306 . . . . 5 (𝑢 {𝑥𝜑}𝑤 ↔ ∀𝑥(𝜑𝑢𝑥𝑤))
18 opex 5321 . . . . . . 7 𝑤, 𝑣⟩ ∈ V
1918elintab 4849 . . . . . 6 (⟨𝑤, 𝑣⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
20 df-br 5031 . . . . . 6 (𝑤 {𝑥𝜑}𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ {𝑥𝜑})
21 df-br 5031 . . . . . . . 8 (𝑤𝑥𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ 𝑥)
2221imbi2i 339 . . . . . . 7 ((𝜑𝑤𝑥𝑣) ↔ (𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
2322albii 1821 . . . . . 6 (∀𝑥(𝜑𝑤𝑥𝑣) ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
2419, 20, 233bitr4i 306 . . . . 5 (𝑤 {𝑥𝜑}𝑣 ↔ ∀𝑥(𝜑𝑤𝑥𝑣))
2517, 24anbi12i 629 . . . 4 ((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) ↔ (∀𝑥(𝜑𝑢𝑥𝑤) ∧ ∀𝑥(𝜑𝑤𝑥𝑣)))
26 opex 5321 . . . . . 6 𝑢, 𝑣⟩ ∈ V
2726elintab 4849 . . . . 5 (⟨𝑢, 𝑣⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
28 df-br 5031 . . . . 5 (𝑢 {𝑥𝜑}𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ {𝑥𝜑})
29 df-br 5031 . . . . . . 7 (𝑢𝑥𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑥)
3029imbi2i 339 . . . . . 6 ((𝜑𝑢𝑥𝑣) ↔ (𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
3130albii 1821 . . . . 5 (∀𝑥(𝜑𝑢𝑥𝑣) ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
3227, 28, 313bitr4i 306 . . . 4 (𝑢 {𝑥𝜑}𝑣 ↔ ∀𝑥(𝜑𝑢𝑥𝑣))
3310, 25, 323imtr4i 295 . . 3 ((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣)
3433gen2 1798 . 2 𝑤𝑣((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣)
351, 34mpgbir 1801 1 ( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wcel 2111  {cab 2776  wss 3881  cop 4531   cint 4838   class class class wbr 5030  ccom 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-int 4839  df-br 5031  df-opab 5093  df-xp 5525  df-rel 5526  df-co 5528
This theorem is referenced by:  dfrtrcl5  40327
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