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Theorem dmcosseqOLD 5991
Description: Obsolete version of dmcosseq 5990 as of 23-Jun-2025. (Contributed by NM, 28-May-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
dmcosseqOLD (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)

Proof of Theorem dmcosseqOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmcoss 5988 . . 3 dom (𝐴𝐵) ⊆ dom 𝐵
21a1i 11 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) ⊆ dom 𝐵)
3 ssel 3989 . . . . . . . 8 (ran 𝐵 ⊆ dom 𝐴 → (𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴))
4 vex 3482 . . . . . . . . . . 11 𝑦 ∈ V
54elrn 5907 . . . . . . . . . 10 (𝑦 ∈ ran 𝐵 ↔ ∃𝑥 𝑥𝐵𝑦)
64eldm 5914 . . . . . . . . . 10 (𝑦 ∈ dom 𝐴 ↔ ∃𝑧 𝑦𝐴𝑧)
75, 6imbi12i 350 . . . . . . . . 9 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) ↔ (∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
8 19.8a 2179 . . . . . . . . . . 11 (𝑥𝐵𝑦 → ∃𝑥 𝑥𝐵𝑦)
98imim1i 63 . . . . . . . . . 10 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧))
10 pm3.2 469 . . . . . . . . . . 11 (𝑥𝐵𝑦 → (𝑦𝐴𝑧 → (𝑥𝐵𝑦𝑦𝐴𝑧)))
1110eximdv 1915 . . . . . . . . . 10 (𝑥𝐵𝑦 → (∃𝑧 𝑦𝐴𝑧 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
129, 11sylcom 30 . . . . . . . . 9 ((∃𝑥 𝑥𝐵𝑦 → ∃𝑧 𝑦𝐴𝑧) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
137, 12sylbi 217 . . . . . . . 8 ((𝑦 ∈ ran 𝐵𝑦 ∈ dom 𝐴) → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
143, 13syl 17 . . . . . . 7 (ran 𝐵 ⊆ dom 𝐴 → (𝑥𝐵𝑦 → ∃𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
1514eximdv 1915 . . . . . 6 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧)))
16 excom 2160 . . . . . 6 (∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧) ↔ ∃𝑦𝑧(𝑥𝐵𝑦𝑦𝐴𝑧))
1715, 16imbitrrdi 252 . . . . 5 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧)))
18 vex 3482 . . . . . . 7 𝑥 ∈ V
19 vex 3482 . . . . . . 7 𝑧 ∈ V
2018, 19opelco 5885 . . . . . 6 (⟨𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2120exbii 1845 . . . . 5 (∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵) ↔ ∃𝑧𝑦(𝑥𝐵𝑦𝑦𝐴𝑧))
2217, 21imbitrrdi 252 . . . 4 (ran 𝐵 ⊆ dom 𝐴 → (∃𝑦 𝑥𝐵𝑦 → ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵)))
2318eldm 5914 . . . 4 (𝑥 ∈ dom 𝐵 ↔ ∃𝑦 𝑥𝐵𝑦)
2418eldm2 5915 . . . 4 (𝑥 ∈ dom (𝐴𝐵) ↔ ∃𝑧𝑥, 𝑧⟩ ∈ (𝐴𝐵))
2522, 23, 243imtr4g 296 . . 3 (ran 𝐵 ⊆ dom 𝐴 → (𝑥 ∈ dom 𝐵𝑥 ∈ dom (𝐴𝐵)))
2625ssrdv 4001 . 2 (ran 𝐵 ⊆ dom 𝐴 → dom 𝐵 ⊆ dom (𝐴𝐵))
272, 26eqssd 4013 1 (ran 𝐵 ⊆ dom 𝐴 → dom (𝐴𝐵) = dom 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wcel 2106  wss 3963  cop 4637   class class class wbr 5148  dom cdm 5689  ran crn 5690  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700
This theorem is referenced by: (None)
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