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Theorem ereldm 6149
 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 2107 . . . 4 (𝜑 → (𝑥 ∈ [𝐴]𝑅𝑥 ∈ [𝐵]𝑅))
32exbidv 1706 . . 3 (𝜑 → (∃𝑥 𝑥 ∈ [𝐴]𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅))
4 ecdmn0m 6148 . . 3 (𝐴 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐴]𝑅)
5 ecdmn0m 6148 . . 3 (𝐵 ∈ dom 𝑅 ↔ ∃𝑥 𝑥 ∈ [𝐵]𝑅)
63, 4, 53bitr4g 212 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
7 ereldm.1 . . . 4 (𝜑𝑅 Er 𝑋)
8 erdm 6116 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
97, 8syl 14 . . 3 (𝜑 → dom 𝑅 = 𝑋)
109eleq2d 2107 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐴𝑋))
119eleq2d 2107 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝑋))
126, 10, 113bitr3d 207 1 (𝜑 → (𝐴𝑋𝐵𝑋))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  dom cdm 4345   Er wer 6103  [cec 6104 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-er 6106  df-ec 6108 This theorem is referenced by:  erth  6150  brecop  6196
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