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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.
StepHypRef Expression
1 elcnv2 4890 . . . 4 (y Azw(y = z, w w, z A))
2 eluni2 3895 . . . . . . 7 (w, z Ax A w, z x)
32anbi2i 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))
53, 4bitr4i 243 . . . . 5 ((y = z, w w, z A) ↔ x A (y = z, w w, z x))
652exbii 1583 . . . 4 (zw(y = z, w w, z A) ↔ zwx A (y = z, w w, z x))
7 elcnv2 4890 . . . . . 6 (y xzw(y = z, w w, z x))
87rexbii 2639 . . . . 5 (x A y xx A zw(y = z, w w, z x))
9 rexcom4 2878 . . . . 5 (x A zw(y = z, w w, z x) ↔ zx A w(y = z, w w, z x))
10 rexcom4 2878 . . . . . 6 (x A w(y = z, w w, z x) ↔ wx A (y = z, w w, z x))
1110exbii 1582 . . . . 5 (zx A w(y = z, w w, z x) ↔ zwx A (y = z, w w, z x))
128, 9, 113bitrri 263 . . . 4 (zwx A (y = z, w w, z x) ↔ x A y x)
131, 6, 123bitri 262 . . 3 (y Ax A y x)
14 eliun 3973 . . 3 (y x A xx A y x)
1513, 14bitr4i 243 . 2 (y Ay x A x)
1615eqriv 2350 1 A = x A x
 Colors of variables: wff setvar class 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-1 5  ax-2 6  ax-3 7  ax-mp 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
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