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Theorem iserd 6043
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 2023 . . 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 1736 . . . . 5 (φz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z)))
98alrimiv 1736 . . . 4 (φyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z)))
109alrimiv 1736 . . 3 (φxyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z)))
11 dfer2 6018 . . 3 (𝑅 Er dom 𝑅 ↔ (Rel 𝑅 dom 𝑅 = dom 𝑅 xyz((x𝑅yy𝑅x) ((x𝑅y y𝑅z) → x𝑅z))))
121, 2, 10, 11syl3anbrc 1075 . 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 6037 . . . . . . 7 ((φ x dom 𝑅) → x𝑅x)
1615ex 108 . . . . . 6 (φ → (x dom 𝑅x𝑅x))
17 vex 2538 . . . . . . 7 x V
1817, 17breldm 4466 . . . . . 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 2020 . . 3 (φ → dom 𝑅 = A)
23 ereq2 6025 . . 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 1226   = wceq 1228   wcel 1374   class class class wbr 3738  dom cdm 4272  Rel wrel 4277   Er wer 6014
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-er 6017
This theorem is referenced by:  swoer  6045  eqer  6049  0er  6051  iinerm  6089  erinxp  6091  ecopover  6115  ecopoverg  6118  enq0er  6290
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