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Theorem iseri 8676
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8675, 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 483 . . 3 ((⊤ ∧ 𝑥𝑅𝑦) → 𝑦𝑅𝑥)
5 iseri.3 . . . 4 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
65adantl 483 . . 3 ((⊤ ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
7 iseri.4 . . . 4 (𝑥𝐴𝑥𝑅𝑥)
87a1i 11 . . 3 (⊤ → (𝑥𝐴𝑥𝑅𝑥))
92, 4, 6, 8iserd 8675 . 2 (⊤ → 𝑅 Er 𝐴)
109mptru 1549 1 𝑅 Er 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wtru 1543  wcel 2107   class class class wbr 5106  Rel wrel 5639   Er wer 8646
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-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-er 8649
This theorem is referenced by:  eqer  8684  0er  8686  ecopover  8761  ener  8942  gicer  19067  hmpher  23138  phtpcer  24361  vitalilem1  24975  tgjustf  27418  erclwwlk  28970  erclwwlkn  29019
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