Theorem funex 3614

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 3613. (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	|- ((Fun F /\ dom F e. B) -> F e. V)

Proof of Theorem funex

Step	Hyp	Ref	Expression
1		fnex 3613	. 2 |- ((F Fn dom F /\ dom F e. B) -> F e. V)
2		funfn 3548	. 2 |- (Fun F <-> F Fn dom F)
3	1, 2	sylanb 451	1 |- ((Fun F /\ dom F e. B) -> F e. V)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 960  Vcvv 1814  dom cdm 3176  Fun wfun 3182   Fn wfn 3183

This theorem is referenced by:  opabex 3615  opabex2g 3617  funrnex 3619  oprabex 4025  oprabex2g 4026

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199

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