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Theorem dmcoss 4649
 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 1426 . . . 4 𝑦𝑦 𝑥𝐵𝑦
2 exsimpl 1549 . . . . 5 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
3 vex 2613 . . . . . 6 𝑥 ∈ V
4 vex 2613 . . . . . 6 𝑦 ∈ V
53, 4opelco 4555 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
6 breq2 3809 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝐵𝑦𝑥𝐵𝑧))
76cbvexv 1838 . . . . 5 (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
82, 5, 73imtr4i 199 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
91, 8exlimi 1526 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
103eldm2 4581 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
113eldm 4580 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
129, 10, 113imtr4i 199 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ dom 𝐵)
1312ssriv 3012 1 dom (𝐴𝐵) ⊆ dom 𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 102  ∃wex 1422   ∈ wcel 1434   ⊆ wss 2982  ⟨cop 3419   class class class wbr 3805  dom cdm 4391   ∘ ccom 4395 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-co 4400  df-dm 4401 This theorem is referenced by:  rncoss  4650  dmcosseq  4651  cossxp  4893  funco  4990  cofunexg  5790
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