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Theorem dmcosseq 5419
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 5417 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
21a1i 11 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐵)
3 ssel 3630 . . . . . . . 8 (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴))
4 vex 3234 . . . . . . . . . . 11 𝑦 ∈ V
54elrn 5398 . . . . . . . . . 10 (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
64eldm 5353 . . . . . . . . . 10 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
75, 6imbi12i 339 . . . . . . . . 9 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8 19.8a 2090 . . . . . . . . . . 11 (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
98imim1i 63 . . . . . . . . . 10 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10 pm3.2 462 . . . . . . . . . . 11 (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦𝑦𝐴𝑧)))
1110eximdv 1886 . . . . . . . . . 10 (𝑥𝐵𝑦 → (∃𝑧 𝑦𝐴𝑧 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
129, 11sylcom 30 . . . . . . . . 9 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
137, 12sylbi 207 . . . . . . . 8 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
143, 13syl 17 . . . . . . 7 (ran 𝐵 ⊆ dom 𝐴 → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
1514eximdv 1886 . . . . . 6 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
16 excom 2082 . . . . . 6 (∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) ↔ ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧))
1715, 16syl6ibr 242 . . . . 5 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧)))
18 vex 3234 . . . . . . 7 𝑥 ∈ V
19 vex 3234 . . . . . . 7 𝑧 ∈ V
2018, 19opelco 5326 . . . . . 6 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2120exbii 1814 . . . . 5 (∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2217, 21syl6ibr 242 . . . 4 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵)))
2318eldm 5353 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
2418eldm2 5354 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵))
2522, 23, 243imtr4g 285 . . 3 (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵𝑥 ∈ dom (𝐴𝐵)))
2625ssrdv 3642 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴𝐵))
272, 26eqssd 3653 1 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wss 3607  cop 4216   class class class wbr 4685  dom cdm 5143  ran crn 5144  ccom 5147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154
This theorem is referenced by:  dmcoeq  5420  fnco  6037  dmcoss3  34343  comptiunov2i  38315  dvsinax  40445  hoicvr  41083  fnresfnco  41527
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