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Theorem bdunexb 11811
Description: Bounded version of unexb 4267. (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 3147 . . . 4 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
21eleq1d 2156 . . 3 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ V ↔ (𝐴𝑦) ∈ V))
3 uneq2 3148 . . . 4 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
43eleq1d 2156 . . 3 (𝑦 = 𝐵 → ((𝐴𝑦) ∈ V ↔ (𝐴𝐵) ∈ V))
5 vex 2622 . . . 4 𝑥 ∈ V
6 vex 2622 . . . 4 𝑦 ∈ V
75, 6bj-unex 11810 . . 3 (𝑥𝑦) ∈ V
82, 4, 7vtocl2g 2683 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝐵) ∈ V)
9 ssun1 3163 . . . 4 𝐴 ⊆ (𝐴𝐵)
10 bdunex.bd1 . . . . 5 BOUNDED 𝐴
1110bdssexg 11795 . . . 4 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐴 ∈ V)
129, 11mpan 415 . . 3 ((𝐴𝐵) ∈ V → 𝐴 ∈ V)
13 ssun2 3164 . . . 4 𝐵 ⊆ (𝐴𝐵)
14 bdunex.bd2 . . . . 5 BOUNDED 𝐵
1514bdssexg 11795 . . . 4 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ V) → 𝐵 ∈ V)
1613, 15mpan 415 . . 3 ((𝐴𝐵) ∈ V → 𝐵 ∈ V)
1712, 16jca 300 . 2 ((𝐴𝐵) ∈ V → (𝐴 ∈ V ∧ 𝐵 ∈ V))
188, 17impbii 124 1 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1289  wcel 1438  Vcvv 2619  cun 2997  wss 2999  BOUNDED wbdc 11731
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-pr 4036  ax-un 4260  ax-bd0 11704  ax-bdor 11707  ax-bdex 11710  ax-bdeq 11711  ax-bdel 11712  ax-bdsb 11713  ax-bdsep 11775
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-uni 3654  df-bdc 11732
This theorem is referenced by: (None)
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