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Theorem ereldm 6085
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 → (A 𝑋B 𝑋))

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 2104 . . . 4 (φ → (x [A]𝑅x [B]𝑅))
32exbidv 1703 . . 3 (φ → (x x [A]𝑅x x [B]𝑅))
4 ecdmn0m 6084 . . 3 (A dom 𝑅x x [A]𝑅)
5 ecdmn0m 6084 . . 3 (B dom 𝑅x x [B]𝑅)
63, 4, 53bitr4g 212 . 2 (φ → (A dom 𝑅B dom 𝑅))
7 ereldm.1 . . . 4 (φ𝑅 Er 𝑋)
8 erdm 6052 . . . 4 (𝑅 Er 𝑋 → dom 𝑅 = 𝑋)
97, 8syl 14 . . 3 (φ → dom 𝑅 = 𝑋)
109eleq2d 2104 . 2 (φ → (A dom 𝑅A 𝑋))
119eleq2d 2104 . 2 (φ → (B dom 𝑅B 𝑋))
126, 10, 113bitr3d 207 1 (φ → (A 𝑋B 𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wex 1378   wcel 1390  dom cdm 4288   Er wer 6039  [cec 6040
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-bnd 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-sbc 2759  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-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-er 6042  df-ec 6044
This theorem is referenced by:  erth  6086  brecop  6132
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