Theorem unieq 3740

Description: Equality theorem for class union. Exercise 15 of [TakeutiZaring] p. 18. (Contributed by NM, 10-Aug-1993.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
unieq	|- ( A = B -> U. A = U. B )

Proof of Theorem unieq

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexeq 2625	. . 3 |- ( A = B -> ( E. x e. A y e. x <-> E. x e. B y e. x ) )
2	1	abbidv 2255	. 2 |- ( A = B -> { y | E. x e. A y e. x } = { y | E. x e. B y e. x } )
3		dfuni2 3733	. 2 |- U. A = { y | E. x e. A y e. x }
4		dfuni2 3733	. 2 |- U. B = { y | E. x e. B y e. x }
5	2, 3, 4	3eqtr4g 2195	1 |- ( A = B -> U. A = U. B )

Syntax hints:   -> wi 4   = wceq 1331   {cab 2123   E.wrex 2415   U.cuni 3731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-uni 3732

This theorem is referenced by:  unieqi  3741  unieqd  3742  uniintsnr  3802  iununir  3891  treq  4027  limeq  4294  uniex  4354  uniexg  4356  ordsucunielexmid  4441  onsucuni2  4474  nnpredcl  4531  elvvuni  4598  unielrel  5061  unixp0im  5070  iotass  5100  nnsucuniel  6384  en1bg  6687  omp1eom  6973  ctmlemr  6986  uniopn  12157  istopon  12169  eltg3  12215  tgdom  12230  cldval  12257  ntrfval  12258  clsfval  12259  neifval  12298  tgrest  12327  cnprcl2k  12364  bj-uniex  13104  bj-uniexg  13105  nnsf  13188  peano3nninf  13190  exmidsbthr  13207