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	⊢ 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 5038	. 2 ⊢ dom 𝐴 = {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣}
2		nfcv 2750	. . . . 5 ⊢ Ⅎ𝑦𝑤
3		dfdmf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
4		nfcv 2750	. . . . 5 ⊢ Ⅎ𝑦𝑣
5	2, 3, 4	nfbr 4623	. . . 4 ⊢ Ⅎ𝑦 𝑤𝐴𝑣
6		nfv 1829	. . . 4 ⊢ Ⅎ𝑣 𝑤𝐴𝑦
7		breq2 4581	. . . 4 ⊢ (𝑣 = 𝑦 → (𝑤𝐴𝑣 ↔ 𝑤𝐴𝑦))
8	5, 6, 7	cbvex 2258	. . 3 ⊢ (∃𝑣 𝑤𝐴𝑣 ↔ ∃𝑦 𝑤𝐴𝑦)
9	8	abbii 2725	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑤𝐴𝑣} = {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦}
10		nfcv 2750	. . . . 5 ⊢ Ⅎ𝑥𝑤
11		dfdmf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
12		nfcv 2750	. . . . 5 ⊢ Ⅎ𝑥𝑦
13	10, 11, 12	nfbr 4623	. . . 4 ⊢ Ⅎ𝑥 𝑤𝐴𝑦
14	13	nfex 2138	. . 3 ⊢ Ⅎ𝑥∃𝑦 𝑤𝐴𝑦
15		nfv 1829	. . 3 ⊢ Ⅎ𝑤∃𝑦 𝑥𝐴𝑦
16		breq1 4580	. . . 4 ⊢ (𝑤 = 𝑥 → (𝑤𝐴𝑦 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1836	. . 3 ⊢ (𝑤 = 𝑥 → (∃𝑦 𝑤𝐴𝑦 ↔ ∃𝑦 𝑥𝐴𝑦))
18	14, 15, 17	cbvab 2732	. 2 ⊢ {𝑤 ∣ ∃𝑦 𝑤𝐴𝑦} = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2635	1 ⊢ dom 𝐴 = {𝑥 ∣ ∃𝑦 𝑥𝐴𝑦}

Syntax hints:   = wceq 1474  ∃wex 1694  {cab 2595  Ⅎwnfc 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