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Theorem dmcoeq 4604
 Description: Domain of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
dmcoeq (dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)

Proof of Theorem dmcoeq
StepHypRef Expression
1 eqimss2 2998 . 2 (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2 dmcosseq 4603 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)
31, 2syl 14 1 (dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917  dom cdm 4345  ran crn 4346   ∘ ccom 4349 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-v 2559  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-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356 This theorem is referenced by:  rncoeq  4605  dfdm2  4852  funcocnv2  5151
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