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Theorem dmcoeq 4705
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 3079 . 2 (dom 𝐴 = ran 𝐵 → ran 𝐵 ⊆ dom 𝐴)
2 dmcosseq 4704 . 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 1289  wss 2999  dom cdm 4438  ran crn 4439  ccom 4442
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449
This theorem is referenced by:  rncoeq  4706  dfdm2  4965  funcocnv2  5278
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