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Theorem cnvuni 5795
Description: The converse of a class union is the (indexed) union of the converses of its members. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
cnvuni 𝐴 = 𝑥𝐴 𝑥
Distinct variable group:   𝑥,𝐴

Proof of Theorem cnvuni
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elcnv2 5786 . . . 4 (𝑦 𝐴 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴))
2 eluni2 4843 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥)
32anbi2i 623 . . . . . 6 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥))
4 r19.42v 3279 . . . . . 6 (∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥))
53, 4bitr4i 277 . . . . 5 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
652exbii 1851 . . . 4 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7 elcnv2 5786 . . . . . 6 (𝑦𝑥 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
87rexbii 3181 . . . . 5 (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9 rexcom4 3233 . . . . 5 (∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10 rexcom4 3233 . . . . . 6 (∃𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
1110exbii 1850 . . . . 5 (∃𝑧𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
128, 9, 113bitrri 298 . . . 4 (∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑥𝐴 𝑦𝑥)
131, 6, 123bitri 297 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
14 eliun 4928 . . 3 (𝑦 𝑥𝐴 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1513, 14bitr4i 277 . 2 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
1615eqriv 2735 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wex 1782  wcel 2106  wrex 3065  cop 4567   cuni 4839   ciun 4924  ccnv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-cnv 5597
This theorem is referenced by:  funcnvuni  7778
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