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Theorem dmcoss 5857
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 2153 . . . 4 𝑦𝑦 𝑥𝐵𝑦
2 exsimpl 1876 . . . . 5 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
3 vex 3426 . . . . . 6 𝑥 ∈ V
4 vex 3426 . . . . . 6 𝑦 ∈ V
53, 4opelco 5757 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
6 breq2 5073 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝐵𝑦𝑥𝐵𝑧))
76cbvexvw 2045 . . . . 5 (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
82, 5, 73imtr4i 295 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
91, 8exlimi 2217 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
103eldm2 5787 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
113eldm 5786 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
129, 10, 113imtr4i 295 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ dom 𝐵)
1312ssriv 3921 1 dom (𝐴𝐵) ⊆ dom 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 399  wex 1787  wcel 2112  wss 3882  cop 4563   class class class wbr 5069  dom cdm 5568  ccom 5572
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-rab 3072  df-v 3424  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-br 5070  df-opab 5132  df-co 5577  df-dm 5578
This theorem is referenced by:  rncoss  5858  dmcosseq  5859  cossxp  6152  fvco4i  6833  cofunexg  7743  fin23lem30  9983  wunco  10374  relexpnndm  14634  mvdco  18867  f1omvdconj  18868  znleval  20549  ofco2  21377  tngtopn  23577  xppreima  30731  cycpmrn  31158  relexp0a  41036  dmtrclfvRP  41050
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