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Theorem dmcoeq 5992
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 4055 . 2 (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2 dmcosseq 5990 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)
31, 2syl 17 1 (dom 𝐴 = ran 𝐵 → dom (𝐴𝐵) = dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wss 3963  dom cdm 5689  ran crn 5690  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700
This theorem is referenced by:  rncoeq  5993  dfdm2  6303  funcocnv2  6874
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