Theorem dfrn2 5835

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 5634	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		df-dm 5633	. 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥}
3		vex 3442	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5829	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
6	5	exbii 1848	. . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
7	6	abbii 2796	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
8	1, 2, 7	3eqtri 2756	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1540  ∃wex 1779  {cab 2707   class class class wbr 5095  ^◡ccnv 5622  dom cdm 5623  ran crn 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-cnv 5631  df-dm 5633  df-rn 5634

This theorem is referenced by:  dfrn3  5836  dfdm4  5842  dm0rn0  5871  rnep  5873  dfrnf  5896  dfima2  6017  funcnv3  6556  opabrn  32573  rncossdmcoss  38434