Theorem dfrn2 5754

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 5561	. 2 ⊢ ran 𝐴 = dom ◡𝐴
2		df-dm 5560	. 2 ⊢ dom ◡𝐴 = {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥}
3		vex 3498	. . . . 5 ⊢ 𝑦 ∈ V
4		vex 3498	. . . . 5 ⊢ 𝑥 ∈ V
5	3, 4	brcnv 5748	. . . 4 ⊢ (𝑦◡𝐴𝑥 ↔ 𝑥𝐴𝑦)
6	5	exbii 1844	. . 3 ⊢ (∃𝑥 𝑦◡𝐴𝑥 ↔ ∃𝑥 𝑥𝐴𝑦)
7	6	abbii 2886	. 2 ⊢ {𝑦 ∣ ∃𝑥 𝑦◡𝐴𝑥} = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}
8	1, 2, 7	3eqtri 2848	1 ⊢ ran 𝐴 = {𝑦 ∣ ∃𝑥 𝑥𝐴𝑦}

Syntax hints:   = wceq 1533  ∃wex 1776  {cab 2799   class class class wbr 5059  ^◡ccnv 5549  dom cdm 5550  ran crn 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-cnv 5558  df-dm 5560  df-rn 5561

This theorem is referenced by:  dfrn3  5755  dfdm4  5759  dm0rn0  5790  rnep  5792  dfrnf  5815  dfima2  5926  funcnv3  6419  opabrn  30357  rncossdmcoss  35689