Theorem funrnex 7133

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 6482. (Contributed by NM, 11-Nov-1995.)

Assertion

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

Proof of Theorem funrnex

Step	Hyp	Ref	Expression
1		funex 6482	. . 3 |- ( ( Fun F /\ dom F e. B ) -> F e. _V )
2	1	ex 450	. 2 |- ( Fun F -> ( dom F e. B -> F e. _V ) )
3		rnexg 7098	. 2 |- ( F e. _V -> ran F e. _V )
4	2, 3	syl6com 37	1 |- ( dom F e. B -> ( Fun F -> ran F e. _V ) )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   dom cdm 5114   ran crn 5115   Fun wfun 5882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  zfrep6  7134  fornex  7135  tz7.48-3  7539  inf0  8518  axcc2lem  9258  zorn2lem4  9321  fnct  9359