Theorem dfdmf 5882

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 5664	. 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑦𝑤
3		dfdmf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 5154	. . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1948	. . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 5109	. . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvexv1 2374	. . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
9	8	abbii 2830	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥𝑤
11		dfdmf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
12		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥𝑦
13	10, 11, 12	nfbr 5154	. . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
14	13	nfex 2357	. . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦
15		nfv 1948	. . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦
16		breq1 5108	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1955	. . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
18	14, 15, 17	cbvabw 2834	. 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2790	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

This proof depends on syntax axioms:   = wceq 1569  ∃wex 1811  {cab 2741  Ⅎwnfc 2910   class class class wbr 5105  dom cdm 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-dm 5664

This theorem is used by:  dmopab  5901