Theorem dfdmf 4930

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 4741	. 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑦𝑤
3		dfdmf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 4140	. . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1577	. . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 4097	. . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvex 1804	. . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
9	8	abbii 2347	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑥𝑤
11		dfdmf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
12		nfcv 2375	. . . . 5 ⊢ Ⅎ𝑥𝑦
13	10, 11, 12	nfbr 4140	. . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
14	13	nfex 1686	. . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦
15		nfv 1577	. . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦
16		breq1 4096	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1873	. . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
18	14, 15, 17	cbvab 2356	. 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2256	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

Syntax hints:   = wceq 1398  ∃wex 1541  {cab 2217  Ⅎwnfc 2362   class class class wbr 4093  dom cdm 4731

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-dm 4741

This theorem is referenced by:  dmopab  4948