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Theorem rncoeq 4948
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 4947 . 2  |-  ( dom  `' B  =  ran  `' A  ->  dom  ( `' B  o.  `' A
)  =  dom  `' A )
2 eqcom 2285 . . 3  |-  ( dom 
A  =  ran  B  <->  ran 
B  =  dom  A
)
3 df-rn 4700 . . . 4  |-  ran  B  =  dom  `' B
4 dfdm4 4872 . . . 4  |-  dom  A  =  ran  `' A
53, 4eqeq12i 2296 . . 3  |-  ( ran 
B  =  dom  A  <->  dom  `' B  =  ran  `' A )
62, 5bitri 240 . 2  |-  ( dom 
A  =  ran  B  <->  dom  `' B  =  ran  `' A )
7 df-rn 4700 . . . 4  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
8 cnvco 4865 . . . . 5  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
98dmeqi 4880 . . . 4  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
107, 9eqtri 2303 . . 3  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
11 df-rn 4700 . . 3  |-  ran  A  =  dom  `' A
1210, 11eqeq12i 2296 . 2  |-  ( ran  ( A  o.  B
)  =  ran  A  <->  dom  ( `' B  o.  `' A )  =  dom  `' A )
131, 6, 123imtr4i 257 1  |-  ( dom 
A  =  ran  B  ->  ran  ( A  o.  B )  =  ran  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   `'ccnv 4688   dom cdm 4689   ran crn 4690    o. ccom 4693
This theorem is referenced by:  dfdm2  5204  foco  5461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700
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