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Theorem bdunexb 10399
Description: Bounded version of unexb 4204. (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 3117 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2122 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 3118 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2122 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 2577 . . . 4 𝑥 ∈ V
6 vex 2577 . . . 4 𝑦 ∈ V
75, 6bj-unex 10398 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 2634 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 3133 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 bdunex.bd1 . . . . 5 BOUNDED 𝐴
1110bdssexg 10383 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
129, 11mpan 408 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
13 ssun2 3134 . . . 4 𝐵 ⊆ (𝐴𝐵)
14 bdunex.bd2 . . . . 5 BOUNDED 𝐵
1514bdssexg 10383 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1613, 15mpan 408 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1712, 16jca 294 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
188, 17impbii 121 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wa 101  wb 102   = wceq 1259  wcel 1409  Vcvv 2574  cun 2942  wss 2944  BOUNDED wbdc 10319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-pr 3971  ax-un 4197  ax-bd0 10292  ax-bdor 10295  ax-bdex 10298  ax-bdeq 10299  ax-bdel 10300  ax-bdsb 10301  ax-bdsep 10363
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-sn 3408  df-pr 3409  df-uni 3608  df-bdc 10320
This theorem is referenced by: (None)
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