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Theorem iserd 7921
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 2771 . . 3 (𝜑 → dom 𝑅 = dom 𝑅)
3 iserd.2 . . . . . . . 8 ((𝜑𝑥𝑅𝑦) → 𝑦𝑅𝑥)
43ex 397 . . . . . . 7 (𝜑 → (𝑥𝑅𝑦𝑦𝑅𝑥))
5 iserd.3 . . . . . . . 8 ((𝜑 ∧ (𝑥𝑅𝑦𝑦𝑅𝑧)) → 𝑥𝑅𝑧)
65ex 397 . . . . . . 7 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
74, 6jca 495 . . . . . 6 (𝜑 → ((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
87alrimiv 2006 . . . . 5 (𝜑 → ∀𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
98alrimiv 2006 . . . 4 (𝜑 → ∀𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
109alrimiv 2006 . . 3 (𝜑 → ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
11 dfer2 7896 . . 3 (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 ∧ dom 𝑅 = dom 𝑅 ∧ ∀𝑥𝑦𝑧((𝑥𝑅𝑦𝑦𝑅𝑥) ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
121, 2, 10, 11syl3anbrc 1427 . 2 (𝜑𝑅 Er dom 𝑅)
1312adantr 466 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑅 Er dom 𝑅)
14 simpr 471 . . . . . . . 8 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥 ∈ dom 𝑅)
1513, 14erref 7915 . . . . . . 7 ((𝜑𝑥 ∈ dom 𝑅) → 𝑥𝑅𝑥)
1615ex 397 . . . . . 6 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
17 vex 3352 . . . . . . 7 𝑥 ∈ V
1817, 17breldm 5467 . . . . . 6 (𝑥𝑅𝑥𝑥 ∈ dom 𝑅)
1916, 18impbid1 215 . . . . 5 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝑅𝑥))
20 iserd.4 . . . . 5 (𝜑 → (𝑥𝐴𝑥𝑅𝑥))
2119, 20bitr4d 271 . . . 4 (𝜑 → (𝑥 ∈ dom 𝑅𝑥𝐴))
2221eqrdv 2768 . . 3 (𝜑 → dom 𝑅 = 𝐴)
23 ereq2 7903 . . 3 (dom 𝑅 = 𝐴 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2422, 23syl 17 . 2 (𝜑 → (𝑅 Er dom 𝑅𝑅 Er 𝐴))
2512, 24mpbid 222 1 (𝜑𝑅 Er 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  wal 1628   = wceq 1630  wcel 2144   class class class wbr 4784  dom cdm 5249  Rel wrel 5254   Er wer 7892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-er 7895
This theorem is referenced by:  iseri  7922  iseriALT  7923  swoer  7925  iiner  7970  erinxp  7972  cicer  16672  eqger  17851  gaorber  17947  efgrelexlemb  18369  efgcpbllemb  18374  hmpher  21807  xmeter  22457  ercgrg  25632  metider  30271
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