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Theorem dmcoeq 4547
 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 2992 . 2 (dom A = ran B → ran B ⊆ dom A)
2 dmcosseq 4546 . 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 1242   ⊆ wss 2911  dom cdm 4288  ran crn 4289   ∘ ccom 4292 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299 This theorem is referenced by:  rncoeq  4548  dfdm2  4795  funcocnv2  5094
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