Theorem cnvuni 5886

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 5877	. . . 4 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴))
2		eluni2 4912	. . . . . . 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 3190	. . . . . 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 5877	. . . . . 6 ⊢ (𝑦 ∈ ◡𝑥 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
8	7	rexbii 3094	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
9		rexcom4 3285	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑧∃𝑥 ∈ 𝐴 ∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
10		rexcom4 3285	. . . . . 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 297	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
13	1, 6, 12	3bitri 296	. . 3 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
14		eliun 5001	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
15	13, 14	bitr4i 277	. 2 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥)
16	15	eqriv 2729	1 ⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥

Syntax hints:   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  ∃wrex 3070  ⟨cop 4634  ∪ cuni 4908  ∪ ciun 4997  ^◡ccnv 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-cnv 5684

This theorem is referenced by:  funcnvuni  7921