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Theorem cnvuni 5750
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 5741 . . . 4 (𝑦 𝐴 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴))
2 eluni2 4834 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥)
32anbi2i 624 . . . . . 6 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥))
4 r19.42v 3348 . . . . . 6 (∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥))
53, 4bitr4i 280 . . . . 5 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
652exbii 1843 . . . 4 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7 elcnv2 5741 . . . . . 6 (𝑦𝑥 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
87rexbii 3245 . . . . 5 (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9 rexcom4 3247 . . . . 5 (∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10 rexcom4 3247 . . . . . 6 (∃𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
1110exbii 1842 . . . . 5 (∃𝑧𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
128, 9, 113bitrri 300 . . . 4 (∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑥𝐴 𝑦𝑥)
131, 6, 123bitri 299 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
14 eliun 4914 . . 3 (𝑦 𝑥𝐴 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1513, 14bitr4i 280 . 2 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
1615eqriv 2816 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1531  wex 1774  wcel 2108  wrex 3137  cop 4565   cuni 4830   ciun 4910  ccnv 5547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-cnv 5556
This theorem is referenced by:  funcnvuni  7628
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