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Theorem bdunexb 15150
Description: Bounded version of unexb 4460. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bdunex.bd1 BOUNDED 𝐴
bdunex.bd2 BOUNDED 𝐵
Assertion
Ref Expression
bdunexb ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)

Proof of Theorem bdunexb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uneq1 3297 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2258 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 3298 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2258 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 2755 . . . 4 𝑥 ∈ V
6 vex 2755 . . . 4 𝑦 ∈ V
75, 6bj-unex 15149 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 2816 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 3313 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 bdunex.bd1 . . . . 5 BOUNDED 𝐴
1110bdssexg 15134 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
129, 11mpan 424 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
13 ssun2 3314 . . . 4 𝐵 ⊆ (𝐴𝐵)
14 bdunex.bd2 . . . . 5 BOUNDED 𝐵
1514bdssexg 15134 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1613, 15mpan 424 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1712, 16jca 306 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
188, 17impbii 126 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105   = wceq 1364  wcel 2160  Vcvv 2752  cun 3142  wss 3144  BOUNDED wbdc 15070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-pr 4227  ax-un 4451  ax-bd0 15043  ax-bdor 15046  ax-bdex 15049  ax-bdeq 15050  ax-bdel 15051  ax-bdsb 15052  ax-bdsep 15114
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3613  df-pr 3614  df-uni 3825  df-bdc 15071
This theorem is referenced by: (None)
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