Theorem fnoprvalrn 4033

Description: An operation's value belongs to its range.

Assertion

Ref	Expression
fnoprvalrn	|- ((F Fn (A X. B) /\ C e. A /\ D e. B) -> (CFD) e. ran F)

Proof of Theorem fnoprvalrn

Step	Hyp	Ref	Expression
1		fnfvelrn 3808	. . . 4 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> (F` <.C, D>.) e. ran F)
2		df-opr 3960	. . . 4 |- (CFD) = (F` <.C, D>.)
3	1, 2	syl5eqel 1550	. . 3 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> (CFD) e. ran F)
4		opelxpi 3213	. . 3 |- ((C e. A /\ D e. B) -> <.C, D>. e. (A X. B))
5	3, 4	sylan2 451	. 2 |- ((F Fn (A X. B) /\ (C e. A /\ D e. B)) -> (CFD) e. ran F)
6	5	3impb 828	1 |- ((F Fn (A X. B) /\ C e. A /\ D e. B) -> (CFD) e. ran F)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   e. wcel 957  <.cop 2408   X. cxp 3164  ran crn 3167   Fn wfn 3173  ` cfv 3178  (class class class)co 3958

This theorem is referenced by:  unirnioo 6348  iooretop 7619

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  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  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-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