Theorem fnoprvalrn 4038

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 3813	. . . 4 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> (F` <.C, D>.) e. ran F)
2		df-opr 3965	. . . 4 |- (CFD) = (F` <.C, D>.)
3	1, 2	syl5eqel 1552	. . 3 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> (CFD) e. ran F)
4		opelxpi 3217	. . 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 829	1 |- ((F Fn (A X. B) /\ C e. A /\ D e. B) -> (CFD) e. ran F)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   e. wcel 958  <.cop 2411   X. cxp 3168  ran crn 3171   Fn wfn 3177  ` cfv 3182  (class class class)co 3963

This theorem is referenced by:  unirnioo 6402  iooretop 7659

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  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  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-rex 1650  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-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-opr 3965

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