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Theorem dmco 5343
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 5027 . 2  |-  dom  ( A  o.  B )  =  ran  `' ( A  o.  B )
2 cnvco 5020 . . 3  |-  `' ( A  o.  B )  =  ( `' B  o.  `' A )
32rneqi 5061 . 2  |-  ran  `' ( A  o.  B
)  =  ran  ( `' B  o.  `' A )
4 rnco2 5342 . . 3  |-  ran  ( `' B  o.  `' A )  =  ( `' B " ran  `' A )
5 dfdm4 5027 . . . 4  |-  dom  A  =  ran  `' A
65imaeq2i 5166 . . 3  |-  ( `' B " dom  A
)  =  ( `' B " ran  `' A )
74, 6eqtr4i 2476 . 2  |-  ran  ( `' B  o.  `' A )  =  ( `' B " dom  A
)
81, 3, 73eqtri 2477 1  |-  dom  ( A  o.  B )  =  ( `' B " dom  A )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837    o. ccom 4838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847
This theorem is referenced by:  curry1  6888  curry2  6891  smobeth  9011  hashkf  12517  imasless  15446  ofco2  19476  fcoinver  28214  xppreima  28248  smatrcl  28622
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