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Theorem rncoeq 4816
Description: Range of a composition. (Contributed by NM, 19-Mar-1998.)
Assertion
Ref Expression
rncoeq  |-  ( dom 
A  =  ran  B  ->  ran  ( A  o.  B )  =  ran  A )

Proof of Theorem rncoeq
StepHypRef Expression
1 dmcoeq 4815 . 2  |-  ( dom  `' B  =  ran  `' A  ->  dom  ( `' B  o.  `' A
)  =  dom  `' A )
2 eqcom 2142 . . 3  |-  ( dom 
A  =  ran  B  <->  ran 
B  =  dom  A
)
3 df-rn 4554 . . . 4  |-  ran  B  =  dom  `' B
4 dfdm4 4735 . . . 4  |-  dom  A  =  ran  `' A
53, 4eqeq12i 2154 . . 3  |-  ( ran 
B  =  dom  A  <->  dom  `' B  =  ran  `' A )
62, 5bitri 183 . 2  |-  ( dom 
A  =  ran  B  <->  dom  `' B  =  ran  `' A )
7 df-rn 4554 . . . 4  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
8 cnvco 4728 . . . . 5  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
98dmeqi 4744 . . . 4  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
107, 9eqtri 2161 . . 3  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
11 df-rn 4554 . . 3  |-  ran  A  =  dom  `' A
1210, 11eqeq12i 2154 . 2  |-  ( ran  ( A  o.  B
)  =  ran  A  <->  dom  ( `' B  o.  `' A )  =  dom  `' A )
131, 6, 123imtr4i 200 1  |-  ( dom 
A  =  ran  B  ->  ran  ( A  o.  B )  =  ran  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1332   `'ccnv 4542   dom cdm 4543   ran crn 4544    o. ccom 4547
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2689  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-br 3934  df-opab 3994  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554
This theorem is referenced by:  dfdm2  5077  foco  5359
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