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Theorem dmco 4857
 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 4555 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 4548 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 4590 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 4856 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 4555 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 4694 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2079 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2080 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
 Colors of variables: wff set class Syntax hints:   = wceq 1259  ◡ccnv 4372  dom cdm 4373  ran crn 4374   “ cima 4376   ∘ ccom 4377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386 This theorem is referenced by: (None)
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