Theorem fnrnoprval 4031

Description: The range of an operation expressed as a collection of the operation's values.

Assertion

Ref	Expression
fnrnoprval	|- (F Fn (A X. B) -> ran F = {z | E.x e. A E.y e. B z = (xFy)})

Distinct variable groups:   x,y,z,A   x,B,y,z   x,F,y,z

Proof of Theorem fnrnoprval

Step	Hyp	Ref	Expression
1		fnrnfv 3754	. 2 |- (F Fn (A X. B) -> ran F = {z | E.w e. (A X. B)z = (F` w)})
2		fveq2 3719	. . . . . 6 |- (w = <.x, y>. -> (F` w) = (F` <.x, y>.))
3		df-opr 3960	. . . . . 6 |- (xFy) = (F` <.x, y>.)
4	2, 3	syl6eqr 1523	. . . . 5 |- (w = <.x, y>. -> (F` w) = (xFy))
5	4	eqeq2d 1484	. . . 4 |- (w = <.x, y>. -> (z = (F` w) <-> z = (xFy)))
6	5	rexxp 3215	. . 3 |- (E.w e. (A X. B)z = (F` w) <-> E.x e. A E.y e. B z = (xFy))
7	6	abbii 1573	. 2 |- {z | E.w e. (A X. B)z = (F` w)} = {z | E.x e. A E.y e. B z = (xFy)}
8	1, 7	syl6eq 1521	1 |- (F Fn (A X. B) -> ran F = {z | E.x e. A E.y e. B z = (xFy)})

Syntax hints:   -> wi 3   = wceq 955  {cab 1462  E.wrex 1644  <.cop 2408   X. cxp 3164  ran crn 3167   Fn wfn 3173  ` cfv 3178  (class class class)co 3958

This theorem is referenced by:  oprvalelrn 4034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194  df-opr 3960

Copyright terms: Public domain