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Theorem dmcosseq 4651
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 4649 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
21a1i 9 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐵)
3 ssel 3002 . . . . . . . 8 (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴))
4 vex 2613 . . . . . . . . . . 11 𝑦 ∈ V
54elrn 4625 . . . . . . . . . 10 (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
64eldm 4580 . . . . . . . . . 10 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
75, 6imbi12i 237 . . . . . . . . 9 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8 19.8a 1523 . . . . . . . . . . 11 (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
98imim1i 59 . . . . . . . . . 10 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10 pm3.2 137 . . . . . . . . . . 11 (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦𝑦𝐴𝑧)))
1110eximdv 1803 . . . . . . . . . 10 (𝑥𝐵𝑦 → (∃𝑧 𝑦𝐴𝑧 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
129, 11sylcom 28 . . . . . . . . 9 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
137, 12sylbi 119 . . . . . . . 8 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
143, 13syl 14 . . . . . . 7 (ran 𝐵 ⊆ dom 𝐴 → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
1514eximdv 1803 . . . . . 6 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
16 excom 1595 . . . . . 6 (∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) ↔ ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧))
1715, 16syl6ibr 160 . . . . 5 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧)))
18 vex 2613 . . . . . . 7 𝑥 ∈ V
19 vex 2613 . . . . . . 7 𝑧 ∈ V
2018, 19opelco 4555 . . . . . 6 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2120exbii 1537 . . . . 5 (∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2217, 21syl6ibr 160 . . . 4 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵)))
2318eldm 4580 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
2418eldm2 4581 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵))
2522, 23, 243imtr4g 203 . . 3 (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵𝑥 ∈ dom (𝐴𝐵)))
2625ssrdv 3014 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴𝐵))
272, 26eqssd 3025 1 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wss 2982  cop 3419   class class class wbr 3805  dom cdm 4391  ran crn 4392  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-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402
This theorem is referenced by:  dmcoeq  4652  fnco  5058
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