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Theorem bdunexb 12810
Description: Bounded version of unexb 4323. (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 3189 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2183 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 3190 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2183 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 2660 . . . 4 𝑥 ∈ V
6 vex 2660 . . . 4 𝑦 ∈ V
75, 6bj-unex 12809 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 2721 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 3205 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 bdunex.bd1 . . . . 5 BOUNDED 𝐴
1110bdssexg 12794 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
129, 11mpan 418 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
13 ssun2 3206 . . . 4 𝐵 ⊆ (𝐴𝐵)
14 bdunex.bd2 . . . . 5 BOUNDED 𝐵
1514bdssexg 12794 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1613, 15mpan 418 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1712, 16jca 302 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
188, 17impbii 125 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104   = wceq 1314  wcel 1463  Vcvv 2657  cun 3035  wss 3037  BOUNDED wbdc 12730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-pr 4091  ax-un 4315  ax-bd0 12703  ax-bdor 12706  ax-bdex 12709  ax-bdeq 12710  ax-bdel 12711  ax-bdsb 12712  ax-bdsep 12774
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-pr 3500  df-uni 3703  df-bdc 12731
This theorem is referenced by: (None)
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