Theorem cnvuni 5911

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 5902	. . . 4 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴))
2		eluni2 4935	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥)
3	2	anbi2i 622	. . . . . 6 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥))
4		r19.42v 3197	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥))
5	3, 4	bitr4i 278	. . . . 5 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
6	5	2exbii 1847	. . . 4 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7		elcnv2 5902	. . . . . 6 ⊢ (𝑦 ∈ ◡𝑥 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
8	7	rexbii 3100	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9		rexcom4 3294	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10		rexcom4 3294	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
11	10	exbii 1846	. . . . 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 5019	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
15	13, 14	bitr4i 278	. 2 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥)
16	15	eqriv 2737	1 ⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥

Syntax hints:   ∧ wa 395   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∃wrex 3076  ⟨cop 4654  ∪ cuni 4931  ∪ ciun 5015  ^◡ccnv 5699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-cnv 5708

This theorem is referenced by:  funcnvuni  7972