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Theorem iseri 8751
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. Inference version of iserd 8750, 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 8750 . 2 (⊤ → 𝑅 Er 𝐴)
109mptru 1541 1 𝑅 Er 𝐴
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wtru 1535  wcel 2099   class class class wbr 5148  Rel wrel 5683   Er wer 8721
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-er 8724
This theorem is referenced by:  eqer  8759  0er  8761  ecopover  8839  ener  9021  gicer  19230  hmpher  23687  phtpcer  24920  vitalilem1  25536  tgjustf  28276  erclwwlk  29832  erclwwlkn  29881  gricer  47190
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