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Theorem cnvuni 4464
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.
StepHypRef Expression
1 elcnv2 4456 . . . 4 (y Azw(y = ⟨z, ww, z A))
2 eluni2 3575 . . . . . . 7 (⟨w, z Ax Aw, z x)
32anbi2i 430 . . . . . 6 ((y = ⟨z, ww, z A) ↔ (y = ⟨z, w x Aw, z x))
4 r19.42v 2461 . . . . . 6 (x A (y = ⟨z, ww, z x) ↔ (y = ⟨z, w x Aw, z x))
53, 4bitr4i 176 . . . . 5 ((y = ⟨z, ww, z A) ↔ x A (y = ⟨z, ww, z x))
652exbii 1494 . . . 4 (zw(y = ⟨z, ww, z A) ↔ zwx A (y = ⟨z, ww, z x))
7 elcnv2 4456 . . . . . 6 (y xzw(y = ⟨z, ww, z x))
87rexbii 2325 . . . . 5 (x A y xx A zw(y = ⟨z, ww, z x))
9 rexcom4 2571 . . . . 5 (x A zw(y = ⟨z, ww, z x) ↔ zx A w(y = ⟨z, ww, z x))
10 rexcom4 2571 . . . . . 6 (x A w(y = ⟨z, ww, z x) ↔ wx A (y = ⟨z, ww, z x))
1110exbii 1493 . . . . 5 (zx A w(y = ⟨z, ww, z x) ↔ zwx A (y = ⟨z, ww, z x))
128, 9, 113bitrri 196 . . . 4 (zwx A (y = ⟨z, ww, z x) ↔ x A y x)
131, 6, 123bitri 195 . . 3 (y Ax A y x)
14 eliun 3652 . . 3 (y x A xx A y x)
1513, 14bitr4i 176 . 2 (y Ay x A x)
1615eqriv 2034 1 A = x A x
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  wrex 2301  cop 3370   cuni 3571   ciun 3648  ccnv 4287
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-cnv 4296
This theorem is referenced by:  funcnvuni  4911
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