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Theorem erref 6157
Description: An equivalence relation is reflexive on its field. Compare Theorem 3M of [Enderton] p. 56. (Contributed by Mario Carneiro, 6-May-2013.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersymb.1 (𝜑𝑅 Er 𝑋)
erref.2 (𝜑𝐴𝑋)
Assertion
Ref Expression
erref (𝜑𝐴𝑅𝐴)

Proof of Theorem erref
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 erref.2 . . . 4 (𝜑𝐴𝑋)
2 ersymb.1 . . . . 5 (𝜑𝑅 Er 𝑋)
3 erdm 6147 . . . . 5 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
42, 3syl 14 . . . 4 (𝜑 → dom 𝑅 = 𝑋)
51, 4eleqtrrd 2133 . . 3 (𝜑𝐴 ∈ dom 𝑅)
6 eldmg 4558 . . . 4 (𝐴𝑋 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
71, 6syl 14 . . 3 (𝜑 → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
85, 7mpbid 139 . 2 (𝜑 → ∃𝑥 𝐴𝑅𝑥)
92adantr 265 . . 3 ((𝜑𝐴𝑅𝑥) → 𝑅 Er 𝑋)
10 simpr 107 . . 3 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝑥)
119, 10, 10ertr4d 6156 . 2 ((𝜑𝐴𝑅𝑥) → 𝐴𝑅𝐴)
128, 11exlimddv 1794 1 (𝜑𝐴𝑅𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409   class class class wbr 3792  dom cdm 4373   Er wer 6134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-er 6137
This theorem is referenced by:  iserd  6163  erth  6181  iinerm  6209  erinxp  6211
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