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Theorem dmcoss 5815
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 2155 . . . 4 𝑦𝑦 𝑥𝐵𝑦
2 exsimpl 1870 . . . . 5 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
3 vex 3474 . . . . . 6 𝑥 ∈ V
4 vex 3474 . . . . . 6 𝑦 ∈ V
53, 4opelco 5715 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
6 breq2 5043 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝐵𝑦𝑥𝐵𝑧))
76cbvexvw 2045 . . . . 5 (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
82, 5, 73imtr4i 295 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
91, 8exlimi 2218 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
103eldm2 5743 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
113eldm 5742 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
129, 10, 113imtr4i 295 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ dom 𝐵)
1312ssriv 3947 1 dom (𝐴𝐵) ⊆ dom 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 399  wex 1781  wcel 2115  wss 3910  cop 4546   class class class wbr 5039  dom cdm 5528  ccom 5532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-co 5537  df-dm 5538
This theorem is referenced by:  rncoss  5816  dmcosseq  5817  cossxp  6096  fvco4i  6735  cofunexg  7625  fin23lem30  9741  wunco  10132  relexpnndm  14379  mvdco  18551  f1omvdconj  18552  znleval  20676  ofco2  21035  tngtopn  23234  xppreima  30380  cycpmrn  30792  relexp0a  40198  dmtrclfvRP  40212
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