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Theorem dmco 5796
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 5463 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 5455 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 5499 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 5795 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 5463 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 5614 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2777 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2778 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1624  ccnv 5257  dom cdm 5258  ran crn 5259  cima 5261  ccom 5262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-xp 5264  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271
This theorem is referenced by:  curry1  7429  curry2  7432  smobeth  9592  hashkf  13305  imasless  16394  ofco2  20451  fcoinver  29717  xppreima  29750  smatrcl  30163  fco3  39912
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