Theorem cnvuni 5874

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 5861	. . . 4 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴))
2		eluni2 4874	. . . . . . 7 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑤, 𝑧⟩ ∈ 𝑥)
3	2	anbi2i 635	. . . . . 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 281	. . . . 5 ⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
6	5	2exbii 1882	. . . 4 ⊢ (∃𝑧∃𝑤(𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝐴) ↔ ∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥))
7		elcnv2 5861	. . . . . 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 1881	. . . . 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 301	. . . 4 ⊢ (∃𝑧∃𝑤∃𝑥 ∈ 𝐴 (𝑦 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑤, 𝑧⟩ ∈ 𝑥) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
13	1, 6, 12	3bitri 300	. . 3 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
14		eliun 4958	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ◡𝑥)
15	13, 14	bitr4i 281	. 2 ⊢ (𝑦 ∈ ◡∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ◡𝑥)
16	15	eqriv 2759	1 ⊢ ◡∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 ◡𝑥

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088  ⟨cop 4593  ∪ cuni 4870  ∪ ciun 4954  ^◡ccnv 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-rab 3415  df-v 3455  df-un 3907  df-in 3909  df-ss 3919  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-cnv 5667

This theorem is used by:  funcnvuni  7932