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Theorem dmcoeq 4811
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 3152 . 2 (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2 dmcosseq 4810 . 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 1331  wss 3071  dom cdm 4539  ran crn 4540  ccom 4543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550
This theorem is referenced by:  rncoeq  4812  dfdm2  5073  funcocnv2  5392
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