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Theorem brtrclfv 14896
Description: Two ways of expressing the transitive closure of a binary relation. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
brtrclfv (𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))
Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   𝑅,𝑟
Allowed substitution hint:   𝑉(𝑟)

Proof of Theorem brtrclfv
StepHypRef Expression
1 trclfv 14894 . . 3 (𝑅𝑉 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
21breqd 5120 . 2 (𝑅𝑉 → (𝐴(t+‘𝑅)𝐵𝐴 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝐵))
3 brintclab 14895 . . 3 (𝐴 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
4 df-br 5110 . . . . 5 (𝐴𝑟𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑟)
54imbi2i 336 . . . 4 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
65albii 1822 . . 3 (∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵) ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
73, 6bitr4i 278 . 2 (𝐴 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵))
82, 7bitrdi 287 1 (𝑅𝑉 → (𝐴(t+‘𝑅)𝐵 ↔ ∀𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → 𝐴𝑟𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wcel 2107  {cab 2710  wss 3914  cop 4596   cint 4911   class class class wbr 5109  ccom 5641  cfv 6500  t+ctcl 14879
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-ne 2941  df-ral 3062  df-rex 3071  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-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  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-iota 6452  df-fun 6502  df-fv 6508  df-trcl 14881
This theorem is referenced by:  brcnvtrclfv  14897  brtrclfvcnv  14898  trclfvcotr  14903
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