Theorem funrnex 3605

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 3600.

Assertion

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

Proof of Theorem funrnex

Step	Hyp	Ref	Expression
1		funex 3600	. . 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 3353	. 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 956  Vcvv 1807  dom cdm 3165  ran crn 3166  Fun wfun 3171

This theorem is referenced by:  zfrep6 3606  fornex 3670  tz7.48-3 3949  inf0 4586  inf3lem7 4599  noinfep 4620  zorn2lem4 4771

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188

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