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Theorem dfdmf 5226
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 5038 . 2 dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2 nfcv 2750 . . . . 5 𝑦𝑤
3 dfdmf.2 . . . . 5 𝑦𝐴
4 nfcv 2750 . . . . 5 𝑦𝑣
52, 3, 4nfbr 4623 . . . 4 𝑦 𝑤𝐴𝑣
6 nfv 1829 . . . 4 𝑣 𝑤𝐴𝑦
7 breq2 4581 . . . 4 (𝑣 = 𝑦 → (𝑤𝐴𝑣𝑤𝐴𝑦))
85, 6, 7cbvex 2258 . . 3 (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
98abbii 2725 . 2 {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10 nfcv 2750 . . . . 5 𝑥𝑤
11 dfdmf.1 . . . . 5 𝑥𝐴
12 nfcv 2750 . . . . 5 𝑥𝑦
1310, 11, 12nfbr 4623 . . . 4 𝑥 𝑤𝐴𝑦
1413nfex 2138 . . 3 𝑥𝑦 𝑤𝐴𝑦
15 nfv 1829 . . 3 𝑤𝑦 𝑥𝐴𝑦
16 breq1 4580 . . . 4 (𝑤 = 𝑥 → (𝑤𝐴𝑦𝑥𝐴𝑦))
1716exbidv 1836 . . 3 (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
1814, 15, 17cbvab 2732 . 2 {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
191, 9, 183eqtri 2635 1 dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  wex 1694  {cab 2595  wnfc 2737   class class class wbr 4577  dom cdm 5028
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-dm 5038
This theorem is referenced by:  dmopab  5244
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