Theorem funex 2741

Description: If the domain of a function exists, so the function. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of fnex 2740. (Note: Any resemblance between F.U.N.E.X. and "Have You Any Eggs" is purely a coincidence originated by Swedish chefs.)

Assertion

Ref	Expression
funex	|- (dom F e. B -> (Fun F -> F e. V))

Proof of Theorem funex

Step	Hyp	Ref	Expression
1		fnex 2740	. 2 |- (dom F e. B -> (F Fn dom F -> F e. V))
2		df-fn 2433	. . 3 |- (F Fn dom F <-> (Fun F /\ dom F = dom F))
3		cleqid 1102	. . 3 |- dom F = dom F
4	2, 3	mpbiranr 548	. 2 |- (F Fn dom F <-> Fun F)
5	1, 4	syl5ibr 182	1 |- (dom F e. B -> (Fun F -> F e. V))

Syntax hints:   -> wi 2   = wceq 1091   e. wcel 1092  Vcvv 1348  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  funopabex 2742  funrnex 2743  oprabex 3044

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433

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