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Theorem dmcoss 4601
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 1385 . . . 4 𝑦𝑦 𝑥𝐵𝑦
2 exsimpl 1508 . . . . 5 (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) → ∃𝑧 𝑥𝐵𝑧)
3 vex 2560 . . . . . 6 𝑥 ∈ V
4 vex 2560 . . . . . 6 𝑦 ∈ V
53, 4opelco 4507 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) ↔ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦))
6 breq2 3768 . . . . . 6 (𝑦 = 𝑧 → (𝑥𝐵𝑦𝑥𝐵𝑧))
76cbvexv 1795 . . . . 5 (∃𝑦 𝑥𝐵𝑦 ↔ ∃𝑧 𝑥𝐵𝑧)
82, 5, 73imtr4i 190 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
91, 8exlimi 1485 . . 3 (∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵) → ∃𝑦 𝑥𝐵𝑦)
103eldm2 4533 . . 3 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑦𝑥, 𝑦⟩ ∈ (𝐴𝐵))
113eldm 4532 . . 3 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
129, 10, 113imtr4i 190 . 2 (𝑥 ∈ dom (𝐴𝐵) → 𝑥 ∈ dom 𝐵)
1312ssriv 2949 1 dom (𝐴𝐵) ⊆ dom 𝐵
Colors of variables: wff set class
Syntax hints:  wa 97  wex 1381  wcel 1393  wss 2917  cop 3378   class class class wbr 3764  dom cdm 4345  ccom 4349
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-co 4354  df-dm 4355
This theorem is referenced by:  rncoss  4602  dmcosseq  4603  cossxp  4843  funco  4940  cofunexg  5738
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