Theorem funimaex 5976

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4771. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 17-Nov-2002.)

Hypothesis

Ref	Expression
zfrep5.1	|- B e. _V

Assertion

Ref	Expression
funimaex	|- ( Fun A -> ( A " B ) e. _V )

Proof of Theorem funimaex

Step	Hyp	Ref	Expression
1		zfrep5.1	. 2 |- B e. _V
2		funimaexg 5975	. 2 |- ( ( Fun A /\ B e. _V ) -> ( A " B ) e. _V )
3	1, 2	mpan2 707	1 |- ( Fun A -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   "cima 5117   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-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

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890

This theorem is referenced by:  isarep2  5978  isofr  6592  isose  6593  f1opw  6889  f1oweALT  7152  tz9.12lem2  8651  hsmexlem4  9251  hsmexlem5  9252  zorn2lem7  9324  uniimadom  9366  zexALT  11396  fbasrn  21688  fnwe2lem2  37621  setrec2fun  42439