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Theorem dfdmf 5910
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 5699 . 2 dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2 nfcv 2903 . . . . 5 𝑦𝑤
3 dfdmf.2 . . . . 5 𝑦𝐴
4 nfcv 2903 . . . . 5 𝑦𝑣
52, 3, 4nfbr 5195 . . . 4 𝑦 𝑤𝐴𝑣
6 nfv 1912 . . . 4 𝑣 𝑤𝐴𝑦
7 breq2 5152 . . . 4 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvexv1 2343 . . 3 (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
98abbii 2807 . 2 {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10 nfcv 2903 . . . . 5 𝑥𝑤
11 dfdmf.1 . . . . 5 𝑥𝐴
12 nfcv 2903 . . . . 5 𝑥𝑦
1310, 11, 12nfbr 5195 . . . 4 𝑥 𝑤𝐴𝑦
1413nfex 2323 . . 3 𝑥𝑦 𝑤𝐴𝑦
15 nfv 1912 . . 3 𝑤𝑦 𝑥𝐴𝑦
16 breq1 5151 . . . 4 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1716exbidv 1919 . . 3 (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
1814, 15, 17cbvabw 2811 . 2 {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
191, 9, 183eqtri 2767 1 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wex 1776  {cab 2712  wnfc 2888   class class class wbr 5148  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-dm 5699
This theorem is referenced by:  dmopab  5929
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