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Theorem funimaex 6411
Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5154. 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 6410 . 2 ((Fun 𝐴𝐵 ∈ V) → (𝐴𝐵) ∈ V)
31, 2mpan2 690 1 (Fun 𝐴 → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2111  Vcvv 3441  cima 5522  Fun wfun 6318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6326
This theorem is referenced by:  isarep2  6413  isofr  7074  isose  7075  f1opw  7381  f1oweALT  7655  tz9.12lem2  9201  hsmexlem4  9840  hsmexlem5  9841  zorn2lem7  9913  uniimadom  9955  zexALT  11989  fbasrn  22489  dimval  31089  dimvalfi  31090  fnwe2lem2  39990  setrec2fun  45217
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