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Theorem dmcoeq 5101
Description: Domain of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
dmcoeq  |-  ( dom 
A  =  ran  B  ->  dom  ( A  o.  B )  =  dom  B )

Proof of Theorem dmcoeq
StepHypRef Expression
1 eqimss2 3365 . 2  |-  ( dom 
A  =  ran  B  ->  ran  B  C_  dom  A )
2 dmcosseq 5100 . 2  |-  ( ran 
B  C_  dom  A  ->  dom  ( A  o.  B
)  =  dom  B
)
31, 2syl 16 1  |-  ( dom 
A  =  ran  B  ->  dom  ( A  o.  B )  =  dom  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    C_ wss 3284   dom cdm 4841   ran crn 4842    o. ccom 4845
This theorem is referenced by:  rncoeq  5102  dfdm2  5364  funcocnv2  5663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852
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