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

Theorem dmcosseq 5633
Description: Domain of a composition. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dmcosseq (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)

Proof of Theorem dmcosseq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 5631 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
21a1i 11 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐵)
3 ssel 3815 . . . . . . . 8 (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴))
4 vex 3401 . . . . . . . . . . 11 𝑦 ∈ V
54elrn 5612 . . . . . . . . . 10 (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
64eldm 5566 . . . . . . . . . 10 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
75, 6imbi12i 342 . . . . . . . . 9 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8 19.8a 2166 . . . . . . . . . . 11 (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
98imim1i 63 . . . . . . . . . 10 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10 pm3.2 463 . . . . . . . . . . 11 (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦𝑦𝐴𝑧)))
1110eximdv 1960 . . . . . . . . . 10 (𝑥𝐵𝑦 → (∃𝑧 𝑦𝐴𝑧 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
129, 11sylcom 30 . . . . . . . . 9 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
137, 12sylbi 209 . . . . . . . 8 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
143, 13syl 17 . . . . . . 7 (ran 𝐵 ⊆ dom 𝐴 → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
1514eximdv 1960 . . . . . 6 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
16 excom 2155 . . . . . 6 (∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) ↔ ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧))
1715, 16syl6ibr 244 . . . . 5 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧)))
18 vex 3401 . . . . . . 7 𝑥 ∈ V
19 vex 3401 . . . . . . 7 𝑧 ∈ V
2018, 19opelco 5539 . . . . . 6 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2120exbii 1892 . . . . 5 (∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2217, 21syl6ibr 244 . . . 4 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵)))
2318eldm 5566 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
2418eldm2 5567 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵))
2522, 23, 243imtr4g 288 . . 3 (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵𝑥 ∈ dom (𝐴𝐵)))
2625ssrdv 3827 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴𝐵))
272, 26eqssd 3838 1 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wex 1823  wcel 2107  wss 3792  cop 4404   class class class wbr 4886  dom cdm 5355  ran crn 5356  ccom 5359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366
This theorem is referenced by:  dmcoeq  5634  fnco  6245  dmcoss3  34833  comptiunov2i  38959  dvsinax  41059  hoicvr  41693  fnresfnco  42109
  Copyright terms: Public domain W3C validator