Theorem unieq 3928

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 2744	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥))
2	1	abbidv 2354	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥})
3		dfuni2 3921	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
4		dfuni2 3921	. 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
5	2, 3, 4	3eqtr4g 2292	1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)

Syntax hints:   → wi 4   = wceq 1398  {cab 2220  ∃wrex 2523  ∪ cuni 3919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-uni 3920

This theorem is referenced by:  unieqi  3929  unieqd  3930  uniintsnr  3990  iununir  4080  treq  4219  limeq  4503  uniex  4563  uniexg  4565  ordsucunielexmid  4658  onsucuni2  4691  nnpredcl  4750  elvvuni  4819  unielrel  5295  unixp0im  5304  iotass  5335  nnsucuniel  6741  en1bg  7053  omp1eom  7399  ctmlemr  7412  nnnninfeq2  7433  uniopn  14992  istopon  15004  eltg3  15048  tgdom  15063  cldval  15090  ntrfval  15091  clsfval  15092  neifval  15131  tgrest  15160  cnprcl2k  15197  bj-uniex  16813  bj-uniexg  16814  nnsf  16909  peano3nninf  16911  exmidsbthr  16929