Theorem dfrn2 5879

Description: Alternate definition of range. Definition 4 of [Suppes] p. 60. (Contributed by NM, 27-Dec-1996.)

Assertion

Ref	Expression
dfrn2	⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem dfrn2

Step	Hyp	Ref	Expression
1		df-rn 5678	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		df-dm 5677	. 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥}
3		vex 3470	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3470	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5873	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
6	5	exbii 1842	. . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
7	6	abbii 2794	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
8	1, 2, 7	3eqtri 2756	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1533  ∃wex 1773  {cab 2701   class class class wbr 5139  ^◡ccnv 5666  dom cdm 5667  ran crn 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-cnv 5675  df-dm 5677  df-rn 5678

This theorem is referenced by:  dfrn3  5880  dfdm4  5886  dm0rn0  5915  rnep  5917  dfrnf  5940  dfima2  6052  funcnv3  6609  opabrn  32336  rncossdmcoss  37829