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Theorem ereldm 7945
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
StepHypRef Expression
1 ereldm.2 . . . 4 (𝜑 → [𝐴]𝑅 = [𝐵]𝑅)
21neeq1d 2979 . . 3 (𝜑 → ([𝐴]𝑅 ≠ ∅ ↔ [𝐵]𝑅 ≠ ∅))
3 ecdmn0 7944 . . 3 (𝐴 ∈ dom 𝑅 ↔ [𝐴]𝑅 ≠ ∅)
4 ecdmn0 7944 . . 3 (𝐵 ∈ dom 𝑅 ↔ [𝐵]𝑅 ≠ ∅)
52, 3, 43bitr4g 303 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐵 ∈ dom 𝑅))
6 ereldm.1 . . . 4 (𝜑𝑅 Er 𝑋)
7 erdm 7909 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
86, 7syl 17 . . 3 (𝜑 → dom 𝑅 = 𝑋)
98eleq2d 2813 . 2 (𝜑 → (𝐴 ∈ dom 𝑅𝐴𝑋))
108eleq2d 2813 . 2 (𝜑 → (𝐵 ∈ dom 𝑅𝐵𝑋))
115, 9, 103bitr3d 298 1 (𝜑 → (𝐴𝑋𝐵𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1620  wcel 2127  wne 2920  c0 4046  dom cdm 5254   Er wer 7896  [cec 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-er 7899  df-ec 7901
This theorem is referenced by:  erth  7946  brecop  7995  eceqoveq  8007
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