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Theorem dmco 5786
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 5453 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 5445 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 5489 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 5785 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 5453 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 5604 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2796 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2797 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  ccnv 5249  dom cdm 5250  ran crn 5251  cima 5253  ccom 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263
This theorem is referenced by:  curry1  7424  curry2  7427  smobeth  9614  hashkf  13323  imasless  16408  ofco2  20475  fcoinver  29756  xppreima  29789  smatrcl  30202  fco3  39935
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