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Theorem cotrintab 38440
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 5649 . 2 (( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑} ↔ ∀𝑢𝑤𝑣((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣))
2 pm3.43 451 . . . . . 6 (((𝜑𝑢𝑥𝑤) ∧ (𝜑𝑤𝑥𝑣)) → (𝜑 → (𝑢𝑥𝑤𝑤𝑥𝑣)))
3 cotrintab.min . . . . . . 7 (𝜑 → (𝑥𝑥) ⊆ 𝑥)
4 cotr 5649 . . . . . . . 8 ((𝑥𝑥) ⊆ 𝑥 ↔ ∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
54biimpi 206 . . . . . . 7 ((𝑥𝑥) ⊆ 𝑥 → ∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
6 2sp 2209 . . . . . . . 8 (∀𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
76sps 2208 . . . . . . 7 (∀𝑢𝑤𝑣((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣) → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
83, 5, 73syl 18 . . . . . 6 (𝜑 → ((𝑢𝑥𝑤𝑤𝑥𝑣) → 𝑢𝑥𝑣))
92, 8sylcom 30 . . . . 5 (((𝜑𝑢𝑥𝑤) ∧ (𝜑𝑤𝑥𝑣)) → (𝜑𝑢𝑥𝑣))
109alanimi 1891 . . . 4 ((∀𝑥(𝜑𝑢𝑥𝑤) ∧ ∀𝑥(𝜑𝑤𝑥𝑣)) → ∀𝑥(𝜑𝑢𝑥𝑣))
11 opex 5060 . . . . . . 7 𝑢, 𝑤⟩ ∈ V
1211elintab 4620 . . . . . 6 (⟨𝑢, 𝑤⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
13 df-br 4785 . . . . . 6 (𝑢 {𝑥𝜑}𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ {𝑥𝜑})
14 df-br 4785 . . . . . . . 8 (𝑢𝑥𝑤 ↔ ⟨𝑢, 𝑤⟩ ∈ 𝑥)
1514imbi2i 325 . . . . . . 7 ((𝜑𝑢𝑥𝑤) ↔ (𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
1615albii 1894 . . . . . 6 (∀𝑥(𝜑𝑢𝑥𝑤) ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑤⟩ ∈ 𝑥))
1712, 13, 163bitr4i 292 . . . . 5 (𝑢 {𝑥𝜑}𝑤 ↔ ∀𝑥(𝜑𝑢𝑥𝑤))
18 opex 5060 . . . . . . 7 𝑤, 𝑣⟩ ∈ V
1918elintab 4620 . . . . . 6 (⟨𝑤, 𝑣⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
20 df-br 4785 . . . . . 6 (𝑤 {𝑥𝜑}𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ {𝑥𝜑})
21 df-br 4785 . . . . . . . 8 (𝑤𝑥𝑣 ↔ ⟨𝑤, 𝑣⟩ ∈ 𝑥)
2221imbi2i 325 . . . . . . 7 ((𝜑𝑤𝑥𝑣) ↔ (𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
2322albii 1894 . . . . . 6 (∀𝑥(𝜑𝑤𝑥𝑣) ↔ ∀𝑥(𝜑 → ⟨𝑤, 𝑣⟩ ∈ 𝑥))
2419, 20, 233bitr4i 292 . . . . 5 (𝑤 {𝑥𝜑}𝑣 ↔ ∀𝑥(𝜑𝑤𝑥𝑣))
2517, 24anbi12i 604 . . . 4 ((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) ↔ (∀𝑥(𝜑𝑢𝑥𝑤) ∧ ∀𝑥(𝜑𝑤𝑥𝑣)))
26 opex 5060 . . . . . 6 𝑢, 𝑣⟩ ∈ V
2726elintab 4620 . . . . 5 (⟨𝑢, 𝑣⟩ ∈ {𝑥𝜑} ↔ ∀𝑥(𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
28 df-br 4785 . . . . 5 (𝑢 {𝑥𝜑}𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ {𝑥𝜑})
29 df-br 4785 . . . . . . 7 (𝑢𝑥𝑣 ↔ ⟨𝑢, 𝑣⟩ ∈ 𝑥)
3029imbi2i 325 . . . . . 6 ((𝜑𝑢𝑥𝑣) ↔ (𝜑 → ⟨𝑢, 𝑣⟩ ∈ 𝑥))
3130albii 1894 . . . . 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 1870 . 2 𝑤𝑣((𝑢 {𝑥𝜑}𝑤𝑤 {𝑥𝜑}𝑣) → 𝑢 {𝑥𝜑}𝑣)
351, 34mpgbir 1873 1 ( {𝑥𝜑} ∘ {𝑥𝜑}) ⊆ {𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1628  wcel 2144  {cab 2756  wss 3721  cop 4320   cint 4609   class class class wbr 4784  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-int 4610  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-co 5258
This theorem is referenced by:  dfrtrcl5  38455
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