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Theorem dmcoeq 5309
 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 3621 . 2 (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2 dmcosseq 5308 . 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 1475   ⊆ wss 3540  dom cdm 5038  ran crn 5039   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049 This theorem is referenced by:  rncoeq  5310  dfdm2  5584  funcocnv2  6074
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