Theorem dfrn3 2524

Description: Alternate definition of range. Definition 6.5(2) of [TakeutiZaring] p. 24.

Assertion

Ref	Expression
dfrn3	⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}

Distinct variable group(s):   x,y,A

Proof of Theorem dfrn3

Step	Hyp	Ref	Expression
1		dfrn2 2523	. 2 ⊢ ran A = {y∣∃x xAy}
2		df-br 2063	. . . 4 ⊢ (xAy ↔ ⟨x, y⟩ ∈ A)
3	2	biex 733	. . 3 ⊢ (∃x xAy ↔ ∃x⟨x, y⟩ ∈ A)
4	3	biabi 1181	. 2 ⊢ {y∣∃x xAy} = {y∣∃x⟨x, y⟩ ∈ A}
5	1, 4	eqtr 1119	1 ⊢ ran A = {y∣∃x⟨x, y⟩ ∈ A}

Syntax hints:  ∃wex 678  {cab 1090   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  ran crn 2411

This theorem is referenced by:  dfrnf 2561  elrn 2562  rnexg 2569  dfima2 2604  imadmrn 2610  imassrn 2611  fniunfv 2860

This theorem was proved from axioms:  ax-1 3  ax-2 4   5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426  df-dm 2428  df-rn 2429

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