ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rncoeq Unicode version

Theorem rncoeq 4653
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 4652 . 2  |-  ( dom  `' B  =  ran  `' A  ->  dom  ( `' B  o.  `' A
)  =  dom  `' A )
2 eqcom 2085 . . 3  |-  ( dom 
A  =  ran  B  <->  ran 
B  =  dom  A
)
3 df-rn 4402 . . . 4  |-  ran  B  =  dom  `' B
4 dfdm4 4575 . . . 4  |-  dom  A  =  ran  `' A
53, 4eqeq12i 2096 . . 3  |-  ( ran 
B  =  dom  A  <->  dom  `' B  =  ran  `' A )
62, 5bitri 182 . 2  |-  ( dom 
A  =  ran  B  <->  dom  `' B  =  ran  `' A )
7 df-rn 4402 . . . 4  |-  ran  ( A  o.  B )  =  dom  `' ( A  o.  B )
8 cnvco 4568 . . . . 5  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
98dmeqi 4584 . . . 4  |-  dom  `' ( A  o.  B
)  =  dom  ( `' B  o.  `' A )
107, 9eqtri 2103 . . 3  |-  ran  ( A  o.  B )  =  dom  ( `' B  o.  `' A )
11 df-rn 4402 . . 3  |-  ran  A  =  dom  `' A
1210, 11eqeq12i 2096 . 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 1285   `'ccnv 4390   dom cdm 4391   ran crn 4392    o. ccom 4395
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402
This theorem is referenced by:  dfdm2  4902  foco  5167
  Copyright terms: Public domain W3C validator