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Theorem dmeq 4603
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 4602 . . 3 (𝐴𝐵 → dom 𝐴 ⊆ dom 𝐵)
2 dmss 4602 . . 3 (𝐵𝐴 → dom 𝐵 ⊆ dom 𝐴)
31, 2anim12i 331 . 2 ((𝐴𝐵𝐵𝐴) → (dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴))
4 eqss 3029 . 2 (𝐴 = 𝐵 ↔ (𝐴𝐵𝐵𝐴))
5 eqss 3029 . 2 (dom 𝐴 = dom 𝐵 ↔ (dom 𝐴 ⊆ dom 𝐵 ∧ dom 𝐵 ⊆ dom 𝐴))
63, 4, 53imtr4i 199 1 (𝐴 = 𝐵 → dom 𝐴 = dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wss 2988  dom cdm 4410
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-dm 4420
This theorem is referenced by:  dmeqi  4604  dmeqd  4605  xpid11m  4625  fneq1  5064  eqfnfv2  5355  offval  5814  ofrfval  5815  offval3  5856  smoeq  6003  tfrlemi14d  6046  tfr1onlemres  6062  tfrcllemres  6075  rdgivallem  6094  rdgon  6099  rdg0  6100  frec0g  6110  freccllem  6115  frecfcllem  6117  frecsuclem  6119  frecsuc  6120  ereq1  6245  fundmeng  6470
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