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Theorem iserd 8772
Description: A reflexive, symmetric, transitive relation is an equivalence relation on its domain. (Contributed by Mario Carneiro, 9-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
iserd.1 (𝜑 → Rel 𝑅)
iserd.2 ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)
iserd.3 ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
iserd.4 (𝜑 → (𝑥𝐴𝑥𝑅𝑥))
Assertion
Ref Expression
iserd (𝜑𝑅 Er 𝐴)
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴   𝜑,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧)

Proof of Theorem iserd
StepHypRef Expression
1 iserd.1 . . 3 (𝜑 → Rel 𝑅)
2 eqidd 2737 . . 3 (𝜑 → dom 𝑅 = dom 𝑅)
3 iserd.2 . . . . . . . 8 ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)
43ex 412 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦𝑦𝑅𝑥))
5 iserd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
65ex 412 . . . . . . 7 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
74, 6jca 511 . . . . . 6 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
87alrimiv 1926 . . . . 5 (𝜑 → ∀𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
98alrimiv 1926 . . . 4 (𝜑 → ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
109alrimiv 1926 . . 3 (𝜑 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11 dfer2 8747 . . 3 (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 ∧ dom 𝑅 = dom 𝑅 ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
121, 2, 10, 11syl3anbrc 1343 . 2 (𝜑𝑅 Er dom 𝑅)
1312adantr 480 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14 simpr 484 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
1513, 14erref 8766 . . . . . . 7 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
1615ex 412 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
17 vex 3483 . . . . . . 7 𝑥 ∈ V
1817, 17breldm 5918 . . . . . 6 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
1916, 18impbid1 225 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
20 iserd.4 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑅𝑥))
2119, 20bitr4d 282 . . . 4 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝐴))
2221eqrdv 2734 . . 3 (𝜑 → dom 𝑅 = 𝐴)
23 ereq2 8754 . . 3 (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2422, 23syl 17 . 2 (𝜑 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2512, 24mpbid 232 1 (𝜑𝑅 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1537   = wceq 1539  wcel 2107   class class class wbr 5142  dom cdm 5684  Rel wrel 5689   Er wer 8743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-er 8746
This theorem is referenced by:  iseri  8773  iseriALT  8774  swoer  8777  iiner  8830  erinxp  8832  cicer  17851  eqger  19197  gaorber  19327  efgrelexlemb  19769  efgcpbllemb  19774  xmeter  24444  ercgrg  28526  erler  33270  metider  33894  prjsper  42623
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