Theorem cnvuni 4896

Description: The converse of a class union is the (indexed) union of the converses of its members. (Contributed by set.mm contributors, 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 4891	. . . 4 ⊢ (y ∈ ◡∪A ↔ ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A))
2		eluni2 3896	. . . . . . 7 ⊢ (⟨w, z⟩ ∈ ∪A ↔ ∃x ∈ A ⟨w, z⟩ ∈ x)
3	2	anbi2i 675	. . . . . 6 ⊢ ((y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A) ↔ (y = ⟨z, w⟩ ∧ ∃x ∈ A ⟨w, z⟩ ∈ x))
4		r19.42v 2766	. . . . . 6 ⊢ (∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ (y = ⟨z, w⟩ ∧ ∃x ∈ A ⟨w, z⟩ ∈ x))
5	3, 4	bitr4i 243	. . . . 5 ⊢ ((y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A) ↔ ∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
6	5	2exbii 1583	. . . 4 ⊢ (∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A) ↔ ∃z∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
7		elcnv2 4891	. . . . . 6 ⊢ (y ∈ ◡x ↔ ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
8	7	rexbii 2640	. . . . 5 ⊢ (∃x ∈ A y ∈ ◡x ↔ ∃x ∈ A ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
9		rexcom4 2879	. . . . 5 ⊢ (∃x ∈ A ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃z∃x ∈ A ∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
10		rexcom4 2879	. . . . . 6 ⊢ (∃x ∈ A ∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
11	10	exbii 1582	. . . . 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 263	. . . 4 ⊢ (∃z∃w∃x ∈ A (y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃x ∈ A y ∈ ◡x)
13	1, 6, 12	3bitri 262	. . 3 ⊢ (y ∈ ◡∪A ↔ ∃x ∈ A y ∈ ◡x)
14		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A ◡x ↔ ∃x ∈ A y ∈ ◡x)
15	13, 14	bitr4i 243	. 2 ⊢ (y ∈ ◡∪A ↔ y ∈ ∪x ∈ A ◡x)
16	15	eqriv 2350	1 ⊢ ◡∪A = ∪x ∈ A ◡x

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∪cuni 3892  ∪ciun 3970  ⟨cop 4562  ^◡ccnv 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-br 4641  df-cnv 4786

This theorem is referenced by:  funcnvuni  5162