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Theorem dmco 5132
Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
Assertion
Ref Expression
dmco  |-  dom  ( A  o.  B )  =  ( `' B " dom  A )

Proof of Theorem dmco
StepHypRef Expression
1 dfdm4 4814 . 2  |-  dom  ( A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 4807 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 4850 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5131 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `' A )
5 dfdm4 4814 . . . 4  |-  dom  A  =  ran  `' A
65imaeq2i 4963 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `' A )
74, 6eqtr4i 2201 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2202 1  |-  dom  ( A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1353   `'ccnv 4621   dom cdm 4622   ran crn 4623   "cima 4625    o. ccom 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4628  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635
This theorem is referenced by:  casedm  7078  caseinl  7083  caseinr  7084  djudm  7097
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