Theorem dfrnf 5910

Description: Definition of range, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
dfrnf.1	⊢ Ⅎ𝑥𝐴
dfrnf.2	⊢ Ⅎ𝑦𝐴

Assertion

Ref	Expression
dfrnf	⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem dfrnf

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfrn2 5849	. 2 ⊢ ran 𝐴 = {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤}
2		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑣
3		dfrnf.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
4		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑥𝑤
5	2, 3, 4	nfbr 5157	. . . 4 ⊢ Ⅎ𝑥 𝑣𝐴𝑤
6		nfv 1917	. . . 4 ⊢ Ⅎ𝑣 𝑥𝐴𝑤
7		breq1 5113	. . . 4 ⊢ (𝑣 = 𝑥 → (𝑣𝐴𝑤 ↔ 𝑥𝐴𝑤))
8	5, 6, 7	cbvexv1 2338	. . 3 ⊢ (∃𝑣 𝑣𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑤)
9	8	abbii 2801	. 2 ⊢ {𝑤 ∣ ∃𝑣 𝑣𝐴𝑤} = {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤}
10		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑦𝑥
11		dfrnf.2	. . . . 5 ⊢ Ⅎ𝑦𝐴
12		nfcv 2902	. . . . 5 ⊢ Ⅎ𝑦𝑤
13	10, 11, 12	nfbr 5157	. . . 4 ⊢ Ⅎ𝑦 𝑥𝐴𝑤
14	13	nfex 2317	. . 3 ⊢ Ⅎ𝑦∃𝑥 𝑥𝐴𝑤
15		nfv 1917	. . 3 ⊢ Ⅎ𝑤∃𝑥 𝑥𝐴𝑦
16		breq2 5114	. . . 4 ⊢ (𝑤 = 𝑦 → (𝑥𝐴𝑤 ↔ 𝑥𝐴𝑦))
17	16	exbidv 1924	. . 3 ⊢ (𝑤 = 𝑦 → (∃𝑥 𝑥𝐴𝑤 ↔ ∃𝑥 𝑥𝐴𝑦))
18	14, 15, 17	cbvabw 2805	. 2 ⊢ {𝑤 ∣ ∃𝑥 𝑥𝐴𝑤} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
19	1, 9, 18	3eqtri 2763	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1541  ∃wex 1781  {cab 2708  Ⅎwnfc 2882   class class class wbr 5110  ran crn 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-cnv 5646  df-dm 5648  df-rn 5649

This theorem is referenced by:  rnopab  5914