Theorem funimaex 6574

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5221. 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 → (𝐴 “ 𝐵) ∈ V)

Proof of Theorem funimaex

Step	Hyp	Ref	Expression
1		zfrep5.1	. 2 ⊢ 𝐵 ∈ V
2		funimaexg 6573	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 691	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3438   “ cima 5626  Fun wfun 6480

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-fun 6488

This theorem is referenced by:  isarep2  6576  isofr  7283  isose  7284  f1opw  7609  f1oweALT  7914  ttrclse  9642  tz9.12lem2  9703  hsmexlem4  10342  hsmexlem5  10343  zorn2lem7  10415  uniimadom  10457  zexALT  12509  psdmul  22069  fbasrn  23787  oldf  27785  madefi  27845  negsproplem2  27958  precsexlem10  28141  seqsex  28202  noseqex  28206  zsex  28291  dimval  33572  dimvalfi  33573  onvf1odlem4  35078  onvf1od  35079  fnwe2lem2  43024  relpfr  44928  orbitex  44929  permaxpow  44983  permaxun  44985  permac8prim  44988  setrec2fun  49678