Theorem unieq 3944

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 2750	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥 ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥))
2	1	abbidv 2358	. 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥})
3		dfuni2 3937	. 2 ⊢ ∪ 𝐴 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥}
4		dfuni2 3937	. 2 ⊢ ∪ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 ∈ 𝑥}
5	2, 3, 4	3eqtr4g 2296	1 ⊢ (𝐴 = 𝐵 → ∪ 𝐴 = ∪ 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1402  {cab 2224  ∃wrex 2529  ∪ cuni 3935

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3936

This theorem is used by:  unieqi  3945  unieqd  3946  uniintsnr  4006  iununir  4096  treq  4235  limeq  4522  uniex  4583  uniexg  4585  ordsucunielexmid  4678  onsucuni2  4711  nnpredcl  4770  elvvuni  4839  unielrel  5315  unixp0im  5324  iotass  5355  nnsucuniel  6768  en1bg  7087  omp1eom  7435  ctmlemr  7448  nnnninfeq2  7469  uniopn  15102  istopon  15114  eltg3  15158  tgdom  15173  cldval  15200  ntrfval  15201  clsfval  15202  neifval  15241  tgrest  15270  cnprcl2k  15307  bj-uniex  16943  bj-uniexg  16944  nnsf  17048  peano3nninf  17050  exmidsbthr  17068