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Theorem iserd 8675
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 2738 . . 3 (𝜑 → dom 𝑅 = dom 𝑅)
3 iserd.2 . . . . . . . 8 ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)
43ex 414 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦𝑦𝑅𝑥))
5 iserd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
65ex 414 . . . . . . 7 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
74, 6jca 513 . . . . . 6 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
87alrimiv 1931 . . . . 5 (𝜑 → ∀𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
98alrimiv 1931 . . . 4 (𝜑 → ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
109alrimiv 1931 . . 3 (𝜑 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11 dfer2 8650 . . 3 (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 ∧ dom 𝑅 = dom 𝑅 ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
121, 2, 10, 11syl3anbrc 1344 . 2 (𝜑𝑅 Er dom 𝑅)
1312adantr 482 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14 simpr 486 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
1513, 14erref 8669 . . . . . . 7 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
1615ex 414 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
17 vex 3450 . . . . . . 7 𝑥 ∈ V
1817, 17breldm 5865 . . . . . 6 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
1916, 18impbid1 224 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
20 iserd.4 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑅𝑥))
2119, 20bitr4d 282 . . . 4 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝐴))
2221eqrdv 2735 . . 3 (𝜑 → dom 𝑅 = 𝐴)
23 ereq2 8657 . . 3 (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2422, 23syl 17 . 2 (𝜑 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2512, 24mpbid 231 1 (𝜑𝑅 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wcel 2107   class class class wbr 5106  dom cdm 5634  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:  iseri  8676  iseriALT  8677  swoer  8679  iiner  8729  erinxp  8731  cicer  17690  eqger  18981  gaorber  19089  efgrelexlemb  19533  efgcpbllemb  19538  xmeter  23789  ercgrg  27462  metider  32478  prjsper  40949
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