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Theorem dfdisjs5 37224
Description: Alternate definition of the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021.)
Assertion
Ref Expression
dfdisjs5 Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)}
Distinct variable group:   𝑢,𝑟,𝑣

Proof of Theorem dfdisjs5
StepHypRef Expression
1 dfdisjs2 37221 . 2 Disjs = {𝑟 ∈ Rels ∣ ≀ 𝑟 ⊆ I }
2 cosscnvssid5 36990 . . 3 (( ≀ 𝑟 ⊆ I ∧ Rel 𝑟) ↔ (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟))
3 elrelsrelim 37000 . . . . 5 (𝑟 ∈ Rels → Rel 𝑟)
43biantrud 533 . . . 4 (𝑟 ∈ Rels → ( ≀ 𝑟 ⊆ I ↔ ( ≀ 𝑟 ⊆ I ∧ Rel 𝑟)))
53biantrud 533 . . . 4 (𝑟 ∈ Rels → (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ↔ (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟)))
64, 5bibi12d 346 . . 3 (𝑟 ∈ Rels → (( ≀ 𝑟 ⊆ I ↔ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)) ↔ (( ≀ 𝑟 ⊆ I ∧ Rel 𝑟) ↔ (∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅) ∧ Rel 𝑟))))
72, 6mpbiri 258 . 2 (𝑟 ∈ Rels → ( ≀ 𝑟 ⊆ I ↔ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)))
81, 7rabimbieq 36761 1 Disjs = {𝑟 ∈ Rels ∣ ∀𝑢 ∈ dom 𝑟𝑣 ∈ dom 𝑟(𝑢 = 𝑣 ∨ ([𝑢]𝑟 ∩ [𝑣]𝑟) = ∅)}
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3061  {crab 3406  cin 3913  wss 3914  c0 4286   I cid 5534  ccnv 5636  dom cdm 5637  Rel wrel 5642  [cec 8652  ccoss 36684   Rels crels 36686   Disjs cdisjs 36717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rmo 3352  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-ec 8656  df-coss 36923  df-rels 36997  df-ssr 37010  df-cnvrefs 37037  df-cnvrefrels 37038  df-disjss 37215  df-disjs 37216
This theorem is referenced by: (None)
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