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Theorem iserd 6535
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 2171 . . 3 (𝜑 → dom 𝑅 = dom 𝑅)
3 iserd.2 . . . . . . . 8 ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)
43ex 114 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦𝑦𝑅𝑥))
5 iserd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
65ex 114 . . . . . . 7 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
74, 6jca 304 . . . . . 6 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
87alrimiv 1867 . . . . 5 (𝜑 → ∀𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
98alrimiv 1867 . . . 4 (𝜑 → ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
109alrimiv 1867 . . 3 (𝜑 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11 dfer2 6510 . . 3 (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 ∧ dom 𝑅 = dom 𝑅 ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
121, 2, 10, 11syl3anbrc 1176 . 2 (𝜑𝑅 Er dom 𝑅)
1312adantr 274 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14 simpr 109 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
1513, 14erref 6529 . . . . . . 7 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
1615ex 114 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
17 vex 2733 . . . . . . 7 𝑥 ∈ V
1817, 17breldm 4813 . . . . . 6 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
1916, 18impbid1 141 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
20 iserd.4 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑅𝑥))
2119, 20bitr4d 190 . . . 4 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝐴))
2221eqrdv 2168 . . 3 (𝜑 → dom 𝑅 = 𝐴)
23 ereq2 6517 . . 3 (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2422, 23syl 14 . 2 (𝜑 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2512, 24mpbid 146 1 (𝜑𝑅 Er 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1346   = wceq 1348  wcel 2141   class class class wbr 3987  dom cdm 4609  Rel wrel 4614   Er wer 6506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-er 6509
This theorem is referenced by:  swoer  6537  eqer  6541  0er  6543  iinerm  6581  erinxp  6583  ecopover  6607  ecopoverg  6610  ener  6753  enq0er  7384  xmeter  13189
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