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Theorem dmco 5179
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 4871 . 2  |-  dom  (  A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 4864 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 4904 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5178 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `'  A )
5 dfdm4 4871 . . . 4  |-  dom  A  =  ran  `'  A
65imaeq2i 5009 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `'  A )
74, 6eqtr4i 2307 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2308 1  |-  dom  (  A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1624   `'ccnv 4687   dom cdm 4688   ran crn 4689   "cima 4691    o. ccom 4692
This theorem is referenced by:  curry1  6171  curry2  6174  smobeth  8203  hashkf  11333  imasless  13436  domrancur1b  24599  domrancur1c  24601
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701
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