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Theorem dmcoss 4544
 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 dom (AB) ⊆ dom B

Proof of Theorem dmcoss
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfe1 1382 . . . 4 yy xBy
2 exsimpl 1505 . . . . 5 (z(xBz zAy) → z xBz)
3 vex 2554 . . . . . 6 x V
4 vex 2554 . . . . . 6 y V
53, 4opelco 4450 . . . . 5 (⟨x, y (AB) ↔ z(xBz zAy))
6 breq2 3759 . . . . . 6 (y = z → (xByxBz))
76cbvexv 1792 . . . . 5 (y xByz xBz)
82, 5, 73imtr4i 190 . . . 4 (⟨x, y (AB) → y xBy)
91, 8exlimi 1482 . . 3 (yx, y (AB) → y xBy)
103eldm2 4476 . . 3 (x dom (AB) ↔ yx, y (AB))
113eldm 4475 . . 3 (x dom By xBy)
129, 10, 113imtr4i 190 . 2 (x dom (AB) → x dom B)
1312ssriv 2943 1 dom (AB) ⊆ dom B
 Colors of variables: wff set class Syntax hints:   ∧ wa 97  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  dom cdm 4288   ∘ ccom 4292 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-co 4297  df-dm 4298 This theorem is referenced by:  rncoss  4545  dmcosseq  4546  cossxp  4786  funco  4883  cofunexg  5680
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