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 = ∪ 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 5786	. . . 4 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴))
2		eluni2 4843	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥)
3	2	anbi2i 623	. . . . . 6 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥))
4		r19.42v 3279	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥))
5	3, 4	bitr4i 277	. . . . 5 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
6	5	2exbii 1851	. . . 4 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7		elcnv2 5786	. . . . . 6 ⊢ (𝑦 ∈ ◡𝑥 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
8	7	rexbii 3181	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9		rexcom4 3233	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10		rexcom4 3233	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
11	10	exbii 1850	. . . . 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 298	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
13	1, 6, 12	3bitri 297	. . 3 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
14		eliun 4928	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
15	13, 14	bitr4i 277	. 2 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥)
16	15	eqriv 2735	1 ⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥

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