Theorem unieq 3798

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 → ∪ 𝐴 = ∪ 𝐵)

Proof of Theorem unieq

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

Step	Hyp	Ref	Expression
1		rexeq 2662	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥))
2	1	abbidv 2284	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥})
3		dfuni2 3791	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
4		dfuni2 3791	. 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
5	2, 3, 4	3eqtr4g 2224	1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1343  {cab 2151  ∃wrex 2445  ∪ cuni 3789

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-uni 3790

This theorem is referenced by:  unieqi  3799  unieqd  3800  uniintsnr  3860  iununir  3949  treq  4086  limeq  4355  uniex  4415  uniexg  4417  ordsucunielexmid  4508  onsucuni2  4541  nnpredcl  4600  elvvuni  4668  unielrel  5131  unixp0im  5140  iotass  5170  nnsucuniel  6463  en1bg  6766  omp1eom  7060  ctmlemr  7073  nnnninfeq2  7093  uniopn  12639  istopon  12651  eltg3  12697  tgdom  12712  cldval  12739  ntrfval  12740  clsfval  12741  neifval  12780  tgrest  12809  cnprcl2k  12846  bj-uniex  13799  bj-uniexg  13800  nnsf  13885  peano3nninf  13887  exmidsbthr  13902