MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmcoss Structured version   Visualization version   GIF version

Theorem dmcoss 5988
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 2148 . . . 4 𝑦𝑦 𝑥𝐵𝑦
2 exsimpl 1866 . . . . 5 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
3 vex 3482 . . . . . 6 𝑥 ∈ V
4 vex 3482 . . . . . 6 𝑦 ∈ V
53, 4opelco 5885 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
6 breq2 5152 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝐵𝑦𝑥𝐵𝑧))
76cbvexvw 2034 . . . . 5 (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
82, 5, 73imtr4i 292 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
91, 8exlimi 2215 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
103eldm2 5915 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
113eldm 5914 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
129, 10, 113imtr4i 292 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ dom 𝐵)
1312ssriv 3999 1 dom (𝐴𝐵) ⊆ dom 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 395  wex 1776  wcel 2106  wss 3963  cop 4637   class class class wbr 5148  dom cdm 5689  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-co 5698  df-dm 5699
This theorem is referenced by:  rncoss  5989  dmcosseq  5990  dmcosseqOLD  5991  cossxp  6294  fvco4i  7010  cofunexg  7972  fin23lem30  10380  wunco  10771  relexpnndm  15077  mvdco  19478  f1omvdconj  19479  znleval  21591  ofco2  22473  tngtopn  24687  xppreima  32662  cycpmrn  33146  relexp0a  43706  dmtrclfvRP  43720
  Copyright terms: Public domain W3C validator