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Theorem dmco 6088
 Description: The domain of a composition. Exercise 27 of [Enderton] p. 53. (Contributed by NM, 4-Feb-2004.)
Assertion
Ref Expression
dmco dom (𝐴𝐵) = (𝐵 “ dom 𝐴)

Proof of Theorem dmco
StepHypRef Expression
1 dfdm4 5740 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 5730 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 5782 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 6087 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 5740 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 5903 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2784 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2785 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538  ◡ccnv 5526  dom cdm 5527  ran crn 5528   “ cima 5530   ∘ ccom 5531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-xp 5533  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540 This theorem is referenced by:  curry1  7809  curry2  7812  smobeth  10051  hashkf  13747  imasless  16876  ofco2  21156  fcoinver  30473  xppreima  30510  smatrcl  31271  fco3  42253
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