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Theorem dmcoss 5267
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 1840 . . . 4
2 exsimpl 1677 . . . . 5
3 vex 3112 . . . . . 6
4 vex 3112 . . . . . 6
53, 4opelco 5179 . . . . 5
6 breq2 4456 . . . . . 6
76cbvexv 2024 . . . . 5
82, 5, 73imtr4i 266 . . . 4
91, 8exlimi 1912 . . 3
103eldm2 5206 . . 3
113eldm 5205 . . 3
129, 10, 113imtr4i 266 . 2
1312ssriv 3507 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  E.wex 1612  e.wcel 1818  C_wss 3475  <.cop 4035   class class class wbr 4452  domcdm 5004  o.ccom 5008
This theorem is referenced by:  rncoss  5268  dmcosseq  5269  cossxp  5535  fvco4i  5951  cofunexg  6764  fin23lem30  8743  wunco  9132  mvdco  16470  f1omvdconj  16471  znleval  18593  ofco2  18953  tngtopn  21164  xppreima  27487  relexpdm  29058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-co 5013  df-dm 5014
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