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Theorem cnvuni 4448
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 4440 . . . 4 (y Azw(y = ⟨z, ww, z A))
2 eluni2 3558 . . . . . . 7 (⟨w, z Ax Aw, z x)
32anbi2i 433 . . . . . 6 ((y = ⟨z, ww, z A) ↔ (y = ⟨z, w x Aw, z x))
4 r19.42v 2445 . . . . . 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 1479 . . . 4 (zw(y = ⟨z, ww, z A) ↔ zwx A (y = ⟨z, ww, z x))
7 elcnv2 4440 . . . . . 6 (y xzw(y = ⟨z, ww, z x))
87rexbii 2309 . . . . 5 (x A y xx A zw(y = ⟨z, ww, z x))
9 rexcom4 2554 . . . . 5 (x A zw(y = ⟨z, ww, z x) ↔ zx A w(y = ⟨z, ww, z x))
10 rexcom4 2554 . . . . . 6 (x A w(y = ⟨z, ww, z x) ↔ wx A (y = ⟨z, ww, z x))
1110exbii 1478 . . . . 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 3635 . . 3 (y x A xx A y x)
1513, 14bitr4i 176 . 2 (y Ay x A x)
1615eqriv 2019 1 A = x A x
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228  wex 1362   wcel 1374  wrex 2285  cop 3353   cuni 3554   ciun 3631  ccnv 4271
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-cnv 4280
This theorem is referenced by:  funcnvuni  4894
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