Theorem cnvuni 2521

Description: The converse of a class union is the (indexed) union of the converses of its members.

Assertion

Ref	Expression
cnvuni	⊢ ◡∪A = ∪x ∈ A ◡x

Distinct variable group(s):   x,A

Proof of Theorem cnvuni

Step	Hyp	Ref	Expression
1		elcnv2 2515	. . . 4 ⊢ (y ∈ ◡∪A ↔ ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A))
2		eluni2 1923	. . . . . . 7 ⊢ (⟨w, z⟩ ∈ ∪A ↔ ∃x ∈ A ⟨w, z⟩ ∈ x)
3	2	anbi2i 367	. . . . . 6 ⊢ ((y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A) ↔ (y = ⟨z, w⟩ ∧ ∃x ∈ A ⟨w, z⟩ ∈ x))
4		r19.42v 1303	. . . . . 6 ⊢ (∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ (y = ⟨z, w⟩ ∧ ∃x ∈ A ⟨w, z⟩ ∈ x))
5	3, 4	bitr4 154	. . . . 5 ⊢ ((y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A) ↔ ∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
6	5	bi2ex 734	. . . 4 ⊢ (∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A) ↔ ∃z∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
7		rexcom4 1361	. . . . . 6 ⊢ (∃x ∈ A ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃z∃x ∈ A ∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
8		rexcom4 1361	. . . . . . 7 ⊢ (∃x ∈ A ∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
9	8	biex 733	. . . . . 6 ⊢ (∃z∃x ∈ A ∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃z∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
10	7, 9	bitr2 152	. . . . 5 ⊢ (∃z∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃x ∈ A ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
11		elcnv2 2515	. . . . . 6 ⊢ (y ∈ ◡x ↔ ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
12	11	birex 1224	. . . . 5 ⊢ (∃x ∈ A y ∈ ◡x ↔ ∃x ∈ A ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
13	10, 12	bitr4 154	. . . 4 ⊢ (∃z∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃x ∈ A y ∈ ◡x)
14	1, 6, 13	3bitr 155	. . 3 ⊢ (y ∈ ◡∪A ↔ ∃x ∈ A y ∈ ◡x)
15		eliun 1998	. . 3 ⊢ (y ∈ ∪x ∈ A ◡x ↔ ∃x ∈ A y ∈ ◡x)
16	14, 15	bitr4 154	. 2 ⊢ (y ∈ ◡∪A ↔ y ∈ ∪x ∈ A ◡x)
17	16	cleqri 1101	1 ⊢ ◡∪A = ∪x ∈ A ◡x

Syntax hints:   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  ⟨cop 1810  ∪cuni 1919  ∪ciun 1994  ^◡ccnv 2409

This theorem is referenced by:  funcnvuni 2706

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-iun 1996  df-br 2063  df-opab 2098  df-cnv 2426

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