Theorem dfrn3 3304

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

Assertion

Ref	Expression
dfrn3	|- ran A = {y | E.x<.x, y>. e. A}

Distinct variable group:   x,y,A

Proof of Theorem dfrn3

Step	Hyp	Ref	Expression
1		dfrn2 3303	. 2 |- ran A = {y | E.x xAy}
2		df-br 2620	. . . 4 |- (xAy <-> <.x, y>. e. A)
3	2	exbii 1051	. . 3 |- (E.x xAy <-> E.x<.x, y>. e. A)
4	3	abbii 1575	. 2 |- {y | E.x xAy} = {y | E.x<.x, y>. e. A}
5	1, 4	eqtr 1495	1 |- ran A = {y | E.x<.x, y>. e. A}

Syntax hints:   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  <.cop 2411   class class class wbr 2619  ran crn 3171

This theorem is referenced by:  dfrnf 3348  elrn2 3349  dfima2 3405  imadmrn 3414  imassrn 3415  fnrnfv 3759

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-cnv 3186  df-dm 3188  df-rn 3189

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