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Theorem iserd 6068
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 ((φ x𝑅y) → y𝑅x)
iserd.3 ((φ (x𝑅y y𝑅z)) → x𝑅z)
iserd.4 (φ → (x Ax𝑅x))
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 2038 . . 3 (φ → dom 𝑅 = dom 𝑅)
3 iserd.2 . . . . . . . 8 ((φ x𝑅y) → y𝑅x)
43ex 108 . . . . . . 7 (φ → (x𝑅yy𝑅x))
5 iserd.3 . . . . . . . 8 ((φ (x𝑅y y𝑅z)) → x𝑅z)
65ex 108 . . . . . . 7 (φ → ((x𝑅y y𝑅z) → x𝑅z))
74, 6jca 290 . . . . . 6 (φ → ((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z)))
87alrimiv 1751 . . . . 5 (φz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z)))
98alrimiv 1751 . . . 4 (φyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z)))
109alrimiv 1751 . . 3 (φxyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z)))
11 dfer2 6043 . . 3 (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 dom 𝑅 = dom 𝑅 xyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z))))
121, 2, 10, 11syl3anbrc 1087 . 2 (φ𝑅 Er dom 𝑅)
1312adantr 261 . . . . . . . 8 ((φ x dom 𝑅) → 𝑅 Er dom 𝑅)
14 simpr 103 . . . . . . . 8 ((φ x dom 𝑅) → x dom 𝑅)
1513, 14erref 6062 . . . . . . 7 ((φ x dom 𝑅) → x𝑅x)
1615ex 108 . . . . . 6 (φ → (x dom 𝑅x𝑅x))
17 vex 2554 . . . . . . 7 x V
1817, 17breldm 4482 . . . . . 6 (x𝑅xx dom 𝑅)
1916, 18impbid1 130 . . . . 5 (φ → (x dom 𝑅x𝑅x))
20 iserd.4 . . . . 5 (φ → (x Ax𝑅x))
2119, 20bitr4d 180 . . . 4 (φ → (x dom 𝑅x A))
2221eqrdv 2035 . . 3 (φ → dom 𝑅 = A)
23 ereq2 6050 . . 3 (dom 𝑅 = A → (𝑅 Er dom 𝑅𝑅 Er A))
2422, 23syl 14 . 2 (φ → (𝑅 Er dom 𝑅𝑅 Er A))
2512, 24mpbid 135 1 (φ𝑅 Er A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242   wcel 1390   class class class wbr 3755  dom cdm 4288  Rel wrel 4293   Er wer 6039
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-er 6042
This theorem is referenced by:  swoer  6070  eqer  6074  0er  6076  iinerm  6114  erinxp  6116  ecopover  6140  ecopoverg  6143  ener  6195  enq0er  6418
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