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Theorem erref 6062
 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 (φA 𝑋)
Assertion
Ref Expression
erref (φA𝑅A)

Proof of Theorem erref
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 erref.2 . . . 4 (φA 𝑋)
2 ersymb.1 . . . . 5 (φ𝑅 Er 𝑋)
3 erdm 6052 . . . . 5 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
42, 3syl 14 . . . 4 (φ → dom 𝑅 = 𝑋)
51, 4eleqtrrd 2114 . . 3 (φA dom 𝑅)
6 eldmg 4473 . . . 4 (A 𝑋 → (A dom 𝑅x A𝑅x))
71, 6syl 14 . . 3 (φ → (A dom 𝑅x A𝑅x))
85, 7mpbid 135 . 2 (φx A𝑅x)
92adantr 261 . . 3 ((φ A𝑅x) → 𝑅 Er 𝑋)
10 simpr 103 . . 3 ((φ A𝑅x) → A𝑅x)
119, 10, 10ertr4d 6061 . 2 ((φ A𝑅x) → A𝑅A)
128, 11exlimddv 1775 1 (φA𝑅A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390   class class class wbr 3755  dom cdm 4288   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:  iserd  6068  erth  6086  iinerm  6114  erinxp  6116
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