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Theorem ereldm 6265
Description: Equality of equivalence classes implies equivalence of domain membership. (Contributed by NM, 28-Jan-1996.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ereldm.1 (𝜑𝑅 Er 𝑋)
ereldm.2 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
Assertion
Ref Expression
ereldm (𝜑 → (𝐴𝑋𝐵𝑋))

Proof of Theorem ereldm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ereldm.2 . . . . 5 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
21eleq2d 2152 . . . 4 (𝜑 → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
32exbidv 1748 . . 3 (𝜑 → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅))
4 ecdmn0m 6264 . . 3 (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
5 ecdmn0m 6264 . . 3 (𝐵 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅)
63, 4, 53bitr4g 221 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
7 ereldm.1 . . . 4 (𝜑𝑅 Er 𝑋)
8 erdm 6232 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
97, 8syl 14 . . 3 (𝜑 → dom 𝑅 = 𝑋)
109eleq2d 2152 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐴𝑋))
119eleq2d 2152 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝑋))
126, 10, 113bitr3d 216 1 (𝜑 → (𝐴𝑋𝐵𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wex 1422  wcel 1434  dom cdm 4401   Er wer 6219  [cec 6220
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-xp 4407  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-er 6222  df-ec 6224
This theorem is referenced by:  erth  6266  brecop  6312
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