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Theorem cnvuni 5899
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 5890 . . . 4 (𝑦 𝐴 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴))
2 eluni2 4915 . . . . . . 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 3188 . . . . . 6 (∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥))
53, 4bitr4i 278 . . . . 5 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
652exbii 1845 . . . 4 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7 elcnv2 5890 . . . . . 6 (𝑦𝑥 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
87rexbii 3091 . . . . 5 (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9 rexcom4 3285 . . . . 5 (∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10 rexcom4 3285 . . . . . 6 (∃𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
1110exbii 1844 . . . . 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 4999 . . 3 (𝑦 𝑥𝐴 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1513, 14bitr4i 278 . 2 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
1615eqriv 2731 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wex 1775  wcel 2105  wrex 3067  cop 4636   cuni 4911   ciun 4995  ccnv 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rex 3068  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-cnv 5696
This theorem is referenced by:  funcnvuni  7954
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