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Theorem dmeq 4563
Description: Equality theorem for domain. (Contributed by NM, 11-Aug-1994.)
Assertion
Ref Expression
dmeq (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)

Proof of Theorem dmeq
StepHypRef Expression
1 dmss 4562 . . 3 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
2 dmss 4562 . . 3 (𝐵𝐴 → dom 𝐵 ⊆ dom 𝐴)
31, 2anim12i 325 . 2 ((𝐴𝐵𝐵𝐴) → (dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴))
4 eqss 2988 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 2988 . 2 (dom 𝐴 = dom 𝐵 ↔ (dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴))
63, 4, 53imtr4i 194 1 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wss 2945  dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-dm 4383
This theorem is referenced by:  dmeqi  4564  dmeqd  4565  xpid11m  4585  fneq1  5015  eqfnfv2  5294  offval  5747  ofrfval  5748  offval3  5789  smoeq  5936  tfrlemi14d  5978  rdgivallem  5999  rdg0  6005  frec0g  6014  frecsuclem3  6021  frecsuc  6022  ereq1  6144  fundmeng  6318
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