Theorem dfrn2 5544

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 5354	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		df-dm 5353	. 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥}
3		vex 3418	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3418	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5538	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
6	5	exbii 1949	. . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
7	6	abbii 2945	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
8	1, 2, 7	3eqtri 2854	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1658  ∃wex 1880  {cab 2812   class class class wbr 4874  ^◡ccnv 5342  dom cdm 5343  ran crn 5344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4875  df-opab 4937  df-cnv 5351  df-dm 5353  df-rn 5354

This theorem is referenced by:  dfrn3  5545  dfdm4  5549  dm0rn0  5575  dfrnf  5598  dfima2  5710  funcnv3  6193  opabrn  29972  rncossdmcoss  34754