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Theorem rncoeq 4676
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 4675 . 2  |-  ( dom  `' B  =  ran  `' A  ->  dom  ( `' B  o.  `' A
)  =  dom  `' A )
2 eqcom 2087 . . 3  |-  ( dom 
A  =  ran  B  <->  ran 
B  =  dom  A
)
3 df-rn 4424 . . . 4  |-  ran  B  =  dom  `' B
4 dfdm4 4598 . . . 4  |-  dom  A  =  ran  `' A
53, 4eqeq12i 2098 . . 3  |-  ( ran 
B  =  dom  A  <->  dom  `' B  =  ran  `' A )
62, 5bitri 182 . 2  |-  ( dom 
A  =  ran  B  <->  dom  `' B  =  ran  `' A )
7 df-rn 4424 . . . 4  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
8 cnvco 4591 . . . . 5  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
98dmeqi 4607 . . . 4  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
107, 9eqtri 2105 . . 3  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
11 df-rn 4424 . . 3  |-  ran  A  =  dom  `' A
1210, 11eqeq12i 2098 . 2  |-  ( ran  ( A  o.  B
)  =  ran  A  <->  dom  ( `' B  o.  `' A )  =  dom  `' A )
131, 6, 123imtr4i 199 1  |-  ( dom 
A  =  ran  B  ->  ran  ( A  o.  B )  =  ran  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   `'ccnv 4412   dom cdm 4413   ran crn 4414    o. ccom 4417
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424
This theorem is referenced by:  dfdm2  4933  foco  5208
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