Theorem fnex 4535

Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of resfunexg 4500. (The proof was shortened by Andrew Salmon, 17-Sep-2011.)

Assertion

Ref	Expression
fnex	|- ((F Fn A /\ A e. B) -> F e. _V)

Proof of Theorem fnex

Step	Hyp	Ref	Expression
1		fnrel 4511	. . 3 |- (F Fn A -> Rel F)
2	1	adantr 425	. 2 |- ((F Fn A /\ A e. B) -> Rel F)
3		resfunexg 4500	. . . . 5 |- ((Fun F /\ dom F e. B) -> (F |` dom F) e. _V)
4		eleq1a 1966	. . . . . 6 |- (A e. B -> (dom F = A -> dom F e. B))
5	4	impcom 378	. . . . 5 |- ((dom F = A /\ A e. B) -> dom F e. B)
6	3, 5	sylan2 500	. . . 4 |- ((Fun F /\ (dom F = A /\ A e. B)) -> (F |` dom F) e. _V)
7	6	anassrs 489	. . 3 |- (((Fun F /\ dom F = A) /\ A e. B) -> (F |` dom F) e. _V)
8		df-fn 4009	. . 3 |- (F Fn A <-> (Fun F /\ dom F = A))
9	7, 8	sylanb 498	. 2 |- ((F Fn A /\ A e. B) -> (F |` dom F) e. _V)
10		resdm 4249	. . . 4 |- (Rel F -> (F |` dom F) = F)
11	10	eleq1d 1963	. . 3 |- (Rel F -> ((F |` dom F) e. _V <-> F e. _V))
12	11	biimpa 460	. 2 |- ((Rel F /\ (F |` dom F) e. _V) -> F e. _V)
13	2, 9, 12	syl11anc 524	1 |- ((F Fn A /\ A e. B) -> F e. _V)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  dom cdm 3986   |` cres 3988  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993

This theorem is referenced by:  funex 4537  fex 4595  tfrlem12 5130  f1oeng 5454  unfilem3 5643  aceq3lem 5894  ac6lem 5916  ser1absdiflem 8181  climaddci 8392  climmulci 8393  caucvg3ai 8424  caucvg3lem 8426  cvgcmp2clem 8442  cvgcmp2clemOLD 8443  geolimilem 8497  fndmeng 13598  trclex 13937  wfrlem15 13971  cur1vald 14547  domrancur1clem 14549  domrancur1c 14550  valcurfn1 14552  seqzp2 14716  mapdiscn 14871  tailf 15633  elstr 16714  elstrdiff 16720

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009

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