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Theorem rncoss 4871
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
rncoss  |-  ran  ( A  o.  B )  C_ 
ran  A

Proof of Theorem rncoss
StepHypRef Expression
1 dmcoss 4870 . 2  |-  dom  ( `' B  o.  `' A )  C_  dom  `' A
2 df-rn 4612 . . 3  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
3 cnvco 4786 . . . 4  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
43dmeqi 4802 . . 3  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
52, 4eqtri 2185 . 2  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
6 df-rn 4612 . 2  |-  ran  A  =  dom  `' A
71, 5, 63sstr4i 3181 1  |-  ran  ( A  o.  B )  C_ 
ran  A
Colors of variables: wff set class
Syntax hints:    C_ wss 3114   `'ccnv 4600   dom cdm 4601   ran crn 4602    o. ccom 4605
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-v 2726  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-br 3980  df-opab 4041  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612
This theorem is referenced by:  cossxp  5123  fco  5350  caseinj  7048  djuinj  7065
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