Theorem dfdmf 5750

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	⊢ Ⅎ𝑥𝐴
dfdmf.2	⊢ Ⅎ𝑦𝐴

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.

Step	Hyp	Ref	Expression
1		df-dm 5546	. 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑦𝑤
3		dfdmf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 5086	. . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1922	. . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 5043	. . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvexv1 2343	. . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
9	8	abbii 2801	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑥𝑤
11		dfdmf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
12		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑥𝑦
13	10, 11, 12	nfbr 5086	. . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
14	13	nfex 2325	. . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦
15		nfv 1922	. . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦
16		breq1 5042	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1929	. . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
18	14, 15, 17	cbvabw 2805	. 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2763	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

Syntax hints:   = wceq 1543  ∃wex 1787  {cab 2714  Ⅎwnfc 2877   class class class wbr 5039  dom cdm 5536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-dm 5546

This theorem is referenced by:  dmopab  5769