ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnvuni GIF version

Theorem cnvuni 4622
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 4614 . . . 4 (𝑦 𝐴 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴))
2 eluni2 3657 . . . . . . 7 (⟨𝑤, 𝑧⟩ ∈ 𝐴 ↔ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥)
32anbi2i 445 . . . . . 6 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥))
4 r19.42v 2524 . . . . . 6 (∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥𝐴𝑤, 𝑧⟩ ∈ 𝑥))
53, 4bitr4i 185 . . . . 5 ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
652exbii 1542 . . . 4 (∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝐴) ↔ ∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7 elcnv2 4614 . . . . . 6 (𝑦𝑥 ↔ ∃𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
87rexbii 2385 . . . . 5 (∃𝑥𝐴 𝑦𝑥 ↔ ∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9 rexcom4 2642 . . . . 5 (∃𝑥𝐴𝑧𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10 rexcom4 2642 . . . . . 6 (∃𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
1110exbii 1541 . . . . 5 (∃𝑧𝑥𝐴𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
128, 9, 113bitrri 205 . . . 4 (∃𝑧𝑤𝑥𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑥𝐴 𝑦𝑥)
131, 6, 123bitri 204 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
14 eliun 3734 . . 3 (𝑦 𝑥𝐴 𝑥 ↔ ∃𝑥𝐴 𝑦𝑥)
1513, 14bitr4i 185 . 2 (𝑦 𝐴𝑦 𝑥𝐴 𝑥)
1615eqriv 2085 1 𝐴 = 𝑥𝐴 𝑥
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1289  wex 1426  wcel 1438  wrex 2360  cop 3449   cuni 3653   ciun 3730  ccnv 4437
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-cnv 4446
This theorem is referenced by:  funcnvuni  5083
  Copyright terms: Public domain W3C validator