Theorem dfrn2 5913

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 5711	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		df-dm 5710	. 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥}
3		vex 3492	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5907	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
6	5	exbii 1846	. . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
7	6	abbii 2812	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
8	1, 2, 7	3eqtri 2772	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1537  ∃wex 1777  {cab 2717   class class class wbr 5166  ^◡ccnv 5699  dom cdm 5700  ran crn 5701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by:  dfrn3  5914  dfdm4  5920  dm0rn0  5949  rnep  5951  dfrnf  5975  dfima2  6091  funcnv3  6648  opabrn  32634  rncossdmcoss  38411