MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfdmf Structured version   Visualization version   GIF version

Theorem dfdmf 5887
Description: Definition of domain, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 8-Mar-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
dfdmf.1 𝑥𝐴
dfdmf.2 𝑦𝐴
Assertion
Ref Expression
dfdmf dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfdmf
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dm 5677 . 2 dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2 nfcv 2895 . . . . 5 𝑦𝑤
3 dfdmf.2 . . . . 5 𝑦𝐴
4 nfcv 2895 . . . . 5 𝑦𝑣
52, 3, 4nfbr 5186 . . . 4 𝑦 𝑤𝐴𝑣
6 nfv 1909 . . . 4 𝑣 𝑤𝐴𝑦
7 breq2 5143 . . . 4 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvexv1 2330 . . 3 (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
98abbii 2794 . 2 {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10 nfcv 2895 . . . . 5 𝑥𝑤
11 dfdmf.1 . . . . 5 𝑥𝐴
12 nfcv 2895 . . . . 5 𝑥𝑦
1310, 11, 12nfbr 5186 . . . 4 𝑥 𝑤𝐴𝑦
1413nfex 2309 . . 3 𝑥𝑦 𝑤𝐴𝑦
15 nfv 1909 . . 3 𝑤𝑦 𝑥𝐴𝑦
16 breq1 5142 . . . 4 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1716exbidv 1916 . . 3 (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
1814, 15, 17cbvabw 2798 . 2 {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
191, 9, 183eqtri 2756 1 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wex 1773  {cab 2701  wnfc 2875   class class class wbr 5139  dom cdm 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-dm 5677
This theorem is referenced by:  dmopab  5906
  Copyright terms: Public domain W3C validator