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Theorem dmcoss 4854
 Description: Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmcoss

Proof of Theorem dmcoss
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfe1 1476 . . . 4
2 exsimpl 1597 . . . . 5
3 vex 2715 . . . . . 6
4 vex 2715 . . . . . 6
53, 4opelco 4757 . . . . 5
6 breq2 3969 . . . . . 6
76cbvexv 1898 . . . . 5
82, 5, 73imtr4i 200 . . . 4
91, 8exlimi 1574 . . 3
103eldm2 4783 . . 3
113eldm 4782 . . 3
129, 10, 113imtr4i 200 . 2
1312ssriv 3132 1
 Colors of variables: wff set class Syntax hints:   wa 103  wex 1472   wcel 2128   wss 3102  cop 3563   class class class wbr 3965   cdm 4585   ccom 4589 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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-co 4594  df-dm 4595 This theorem is referenced by:  rncoss  4855  dmcosseq  4856  cossxp  5107  funco  5209  cofunexg  6056  casefun  7023  djufun  7042  ctssdccl  7049
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