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Theorem iseri 8727
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8726, which avoids the need to provide a "dummy antecedent" 𝜑 if there is no natural one to choose. (Contributed by AV, 30-Apr-2021.)
Hypotheses
Ref Expression
iseri.1 Rel 𝑅
iseri.2 (𝑥𝑅𝑦𝑦𝑅𝑥)
iseri.3 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
iseri.4 (𝑥𝐴𝑥𝑅𝑥)
Assertion
Ref Expression
iseri 𝑅 Er 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴
Allowed substitution hints:   𝐴(𝑦,𝑧)

Proof of Theorem iseri
StepHypRef Expression
1 iseri.1 . . . 4 Rel 𝑅
21a1i 11 . . 3 (⊤ → Rel 𝑅)
3 iseri.2 . . . 4 (𝑥𝑅𝑦𝑦𝑅𝑥)
43adantl 481 . . 3 ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5 iseri.3 . . . 4 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
65adantl 481 . . 3 ((⊤ ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
7 iseri.4 . . . 4 (𝑥𝐴𝑥𝑅𝑥)
87a1i 11 . . 3 (⊤ → (𝑥𝐴𝑥𝑅𝑥))
92, 4, 6, 8iserd 8726 . 2 (⊤ → 𝑅 Er 𝐴)
109mptru 1540 1 𝑅 Er 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wtru 1534  wcel 2098   class class class wbr 5139  Rel wrel 5672   Er wer 8697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-er 8700
This theorem is referenced by:  eqer  8735  0er  8737  ecopover  8812  ener  8994  gicer  19198  hmpher  23632  phtpcer  24865  vitalilem1  25481  tgjustf  28217  erclwwlk  29770  erclwwlkn  29819
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