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Theorem funimaex 5607
Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4514. 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
StepHypRef Expression
1 zfrep5.1 . 2  |-  B  e. 
_V
2 funimaexg 5606 . 2  |-  ( ( Fun  A  /\  B  e.  _V )  ->  ( A " B )  e. 
_V )
31, 2mpan2 671 1  |-  ( Fun 
A  ->  ( A " B )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758   _Vcvv 3078   "cima 4954   Fun wfun 5523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-fun 5531
This theorem is referenced by:  isarep2  5609  isofr  6145  isose  6146  f1opw  6427  f1oweALT  6674  tz9.12lem2  8109  hsmexlem4  8712  hsmexlem5  8713  zorn2lem7  8785  uniimadom  8822  zexALT  10779  fbasrn  19592  fnwe2lem2  29572
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