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Theorem dfdmf 5899
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 5688 . 2 dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2 nfcv 2899 . . . . 5 𝑦𝑤
3 dfdmf.2 . . . . 5 𝑦𝐴
4 nfcv 2899 . . . . 5 𝑦𝑣
52, 3, 4nfbr 5195 . . . 4 𝑦 𝑤𝐴𝑣
6 nfv 1910 . . . 4 𝑣 𝑤𝐴𝑦
7 breq2 5152 . . . 4 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvexv1 2334 . . 3 (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
98abbii 2798 . 2 {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10 nfcv 2899 . . . . 5 𝑥𝑤
11 dfdmf.1 . . . . 5 𝑥𝐴
12 nfcv 2899 . . . . 5 𝑥𝑦
1310, 11, 12nfbr 5195 . . . 4 𝑥 𝑤𝐴𝑦
1413nfex 2313 . . 3 𝑥𝑦 𝑤𝐴𝑦
15 nfv 1910 . . 3 𝑤𝑦 𝑥𝐴𝑦
16 breq1 5151 . . . 4 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1716exbidv 1917 . . 3 (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
1814, 15, 17cbvabw 2802 . 2 {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
191, 9, 183eqtri 2760 1 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wex 1774  {cab 2705  wnfc 2879   class class class wbr 5148  dom cdm 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-dm 5688
This theorem is referenced by:  dmopab  5918
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