Theorem fnoprvalrn 4116

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 3889	. . . 4 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> (F` <.C, D>.) e. ran F)
2		df-opr 4041	. . . 4 |- (CFD) = (F` <.C, D>.)
3	1, 2	syl5eqel 1589	. . 3 |- ((F Fn (A X. B) /\ <.C, D>. e. (A X. B)) -> (CFD) e. ran F)
4		opelxpi 3274	. . 3 |- ((C e. A /\ D e. B) -> <.C, D>. e. (A X. B))
5	3, 4	sylan2 453	. 2 |- ((F Fn (A X. B) /\ (C e. A /\ D e. B)) -> (CFD) e. ran F)
6	5	3impb 832	1 |- ((F Fn (A X. B) /\ C e. A /\ D e. B) -> (CFD) e. ran F)

Syntax hints:   -> wi 3   /\ wa 221   /\ w3a 778   e. wcel 990  <.cop 2456   X. cxp 3223  ran crn 3226   Fn wfn 3232  ` cfv 3237  (class class class)co 4039

This theorem is referenced by:  unirnioo 6462  iooretop 7779

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-fv 3253  df-opr 4041

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