Theorem funrnex 3613

Description: If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 3608.

Assertion

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

Proof of Theorem funrnex

Step	Hyp	Ref	Expression
1		funex 3608	. . 3 |- ((Fun F /\ dom F e. B) -> F e. V)
2	1	ex 373	. 2 |- (Fun F -> (dom F e. B -> F e. V))
3		rnexg 3359	. 2 |- (F e. V -> ran F e. V)
4	2, 3	syl6com 53	1 |- (dom F e. B -> (Fun F -> ran F e. V))

Syntax hints:   -> wi 3   e. wcel 958  Vcvv 1811  dom cdm 3170  ran crn 3171  Fun wfun 3176

This theorem is referenced by:  zfrep6 3614  fornex 3679  tz7.48-3 3958  inf0 4606  inf3lem7 4619  noinfep 4640  zorn2lem4 4791

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-rep 2693  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-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-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193

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