Theorem dfdmf 4822

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 4638	. 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2		nfcv 2319	. . . . 5 ⊢ Ⅎ𝑦𝑤
3		dfdmf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2319	. . . . 5 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 4051	. . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1528	. . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 4009	. . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvex 1756	. . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
9	8	abbii 2293	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10		nfcv 2319	. . . . 5 ⊢ Ⅎ𝑥𝑤
11		dfdmf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
12		nfcv 2319	. . . . 5 ⊢ Ⅎ𝑥𝑦
13	10, 11, 12	nfbr 4051	. . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
14	13	nfex 1637	. . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦
15		nfv 1528	. . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦
16		breq1 4008	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1825	. . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
18	14, 15, 17	cbvab 2301	. 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2202	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

Syntax hints:   = wceq 1353  ∃wex 1492  {cab 2163  Ⅎwnfc 2306   class class class wbr 4005  dom cdm 4628

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-dm 4638

This theorem is referenced by:  dmopab  4840