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Theorem rncoeq 5079
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 5078 . 2  |-  ( dom  `' B  =  ran  `' A  ->  dom  ( `' B  o.  `' A
)  =  dom  `' A )
2 eqcom 2389 . . 3  |-  ( dom 
A  =  ran  B  <->  ran 
B  =  dom  A
)
3 df-rn 4829 . . . 4  |-  ran  B  =  dom  `' B
4 dfdm4 5003 . . . 4  |-  dom  A  =  ran  `' A
53, 4eqeq12i 2400 . . 3  |-  ( ran 
B  =  dom  A  <->  dom  `' B  =  ran  `' A )
62, 5bitri 241 . 2  |-  ( dom 
A  =  ran  B  <->  dom  `' B  =  ran  `' A )
7 df-rn 4829 . . . 4  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
8 cnvco 4996 . . . . 5  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
98dmeqi 5011 . . . 4  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
107, 9eqtri 2407 . . 3  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
11 df-rn 4829 . . 3  |-  ran  A  =  dom  `' A
1210, 11eqeq12i 2400 . 2  |-  ( ran  ( A  o.  B
)  =  ran  A  <->  dom  ( `' B  o.  `' A )  =  dom  `' A )
131, 6, 123imtr4i 258 1  |-  ( dom 
A  =  ran  B  ->  ran  ( A  o.  B )  =  ran  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   `'ccnv 4817   dom cdm 4818   ran crn 4819    o. ccom 4822
This theorem is referenced by:  dfdm2  5341  foco  5603
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829
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