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Theorem dmco 5042
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 4726 . 2 dom (𝐴𝐵) = ran (𝐴𝐵)
2 cnvco 4719 . . 3 (𝐴𝐵) = (𝐵𝐴)
32rneqi 4762 . 2 ran (𝐴𝐵) = ran (𝐵𝐴)
4 rnco2 5041 . . 3 ran (𝐵𝐴) = (𝐵 “ ran 𝐴)
5 dfdm4 4726 . . . 4 dom 𝐴 = ran 𝐴
65imaeq2i 4874 . . 3 (𝐵 “ dom 𝐴) = (𝐵 “ ran 𝐴)
74, 6eqtr4i 2161 . 2 ran (𝐵𝐴) = (𝐵 “ dom 𝐴)
81, 3, 73eqtri 2162 1 dom (𝐴𝐵) = (𝐵 “ dom 𝐴)
Colors of variables: wff set class
Syntax hints:   = wceq 1331  ccnv 4533  dom cdm 4534  ran crn 4535  cima 4537  ccom 4538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547
This theorem is referenced by:  casedm  6964  caseinl  6969  caseinr  6970  djudm  6983
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