Theorem cnvuni 4895

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 4890	. . . 4 ⊢ (y ∈ ◡∪A ↔ ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ ∪A))
2		eluni2 3895	. . . . . . 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 2765	. . . . . 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 4890	. . . . . 6 ⊢ (y ∈ ◡x ↔ ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
8	7	rexbii 2639	. . . . 5 ⊢ (∃x ∈ A y ∈ ◡x ↔ ∃x ∈ A ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
9		rexcom4 2878	. . . . 5 ⊢ (∃x ∈ A ∃z∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x) ↔ ∃z∃x ∈ A ∃w(y = ⟨z, w⟩ ∧ ⟨w, z⟩ ∈ x))
10		rexcom4 2878	. . . . . 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 3973	. . 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 2615  ∪cuni 3891  ∪ciun 3969  ⟨cop 4561  ^◡ccnv 4771

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-iun 3971  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623  df-br 4640  df-cnv 4785

This theorem is referenced by:  funcnvuni  5161