Theorem cnvuni 5864

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 = ∪ 𝑥 ∈ 𝐴 ◡𝑥

Distinct variable group:   𝑥,𝐴

Proof of Theorem cnvuni

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elcnv2 5851	. . . 4 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴))
2		eluni2 4871	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥)
3	2	anbi2i 632	. . . . . 6 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥))
4		r19.42v 3196	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥))
5	3, 4	bitr4i 280	. . . . 5 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
6	5	2exbii 1871	. . . 4 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7		elcnv2 5851	. . . . . 6 ⊢ (𝑦 ∈ ◡𝑥 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
8	7	rexbii 3111	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9		rexcom4 3291	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10		rexcom4 3291	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
11	10	exbii 1870	. . . . 5 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
12	8, 9, 11	3bitrri 300	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
13	1, 6, 12	3bitri 299	. . 3 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
14		eliun 4955	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
15	13, 14	bitr4i 280	. 2 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥)
16	15	eqriv 2761	1 ⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥

Syntax hints:   ∧ wa 399   = wceq 1562  ∃wex 1801   ∈ wcel 2144  ∃wrex 3088  ⟨cop 4590  ∪ cuni 4867  ∪ ciun 4951  ^◡ccnv 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-11 2193  ax-ext 2736  ax-sep 5248  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rex 3089  df-rab 3417  df-v 3458  df-un 3911  df-in 3913  df-ss 3923  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-cnv 5657

This theorem is referenced by:  funcnvuni  7915