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Theorem funimaex 5890
 Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4699. 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 ∈ V
Assertion
Ref Expression
funimaex (Fun 𝐴 → (𝐴𝐵) ∈ V)

Proof of Theorem funimaex
StepHypRef Expression
1 zfrep5.1 . 2 𝐵 ∈ V
2 funimaexg 5889 . 2 ((Fun 𝐴𝐵 ∈ V) → (𝐴𝐵) ∈ V)
31, 2mpan2 703 1 (Fun 𝐴 → (𝐴𝐵) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977  Vcvv 3173   “ cima 5041  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806 This theorem is referenced by:  isarep2  5892  isofr  6492  isose  6493  f1opw  6787  f1oweALT  7043  tz9.12lem2  8534  hsmexlem4  9134  hsmexlem5  9135  zorn2lem7  9207  uniimadom  9245  zexALT  11273  fbasrn  21498  fnwe2lem2  36639  setrec2fun  42238
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