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Theorem dmco 5369
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 5054 . 2  |-  dom  ( A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 5047 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 5087 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5368 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `' A )
5 dfdm4 5054 . . . 4  |-  dom  A  =  ran  `' A
65imaeq2i 5192 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `' A )
74, 6eqtr4i 2458 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2459 1  |-  dom  ( A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff set class
Syntax hints:    = wceq 1652   `'ccnv 4868   dom cdm 4869   ran crn 4870   "cima 4872    o. ccom 4873
This theorem is referenced by:  curry1  6429  curry2  6432  smobeth  8450  hashkf  11608  imasless  13753  xppreima  24047
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882
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