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Theorem ecdmn0m 6267
Description: A representative of an inhabited equivalence class belongs to the domain of the equivalence relation. (Contributed by Jim Kingdon, 21-Aug-2019.)
Assertion
Ref Expression
ecdmn0m (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
Distinct variable groups:   𝑥,𝑅   𝑥,𝐴

Proof of Theorem ecdmn0m
StepHypRef Expression
1 elex 2623 . 2 (𝐴 ∈ dom 𝑅𝐴 ∈ V)
2 ecexr 6230 . . 3 (𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
32exlimiv 1532 . 2 (∃𝑥 𝑥 ∈ [𝐴]𝑅𝐴 ∈ V)
4 eldmg 4592 . . 3 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
5 vex 2617 . . . . 5 𝑥 ∈ V
6 elecg 6263 . . . . 5 ((𝑥 ∈ V ∧ 𝐴 ∈ V) → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
75, 6mpan 415 . . . 4 (𝐴 ∈ V → (𝑥 ∈ [𝐴]𝑅𝐴𝑅𝑥))
87exbidv 1750 . . 3 (𝐴 ∈ V → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝐴𝑅𝑥))
94, 8bitr4d 189 . 2 (𝐴 ∈ V → (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅))
101, 3, 9pm5.21nii 653 1 (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
Colors of variables: wff set class
Syntax hints:  wb 103  wex 1424  wcel 1436  Vcvv 2614   class class class wbr 3814  dom cdm 4404  [cec 6223
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-cnv 4412  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-ec 6227
This theorem is referenced by:  ereldm  6268  elqsn0m  6293  ecelqsdm  6295
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