Theorem funimaex 6605

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5234. 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 6603	. 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 3447   “ cima 5641  Fun wfun 6505

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 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513

This theorem is referenced by:  isarep2  6608  isofr  7317  isose  7318  f1opw  7645  f1oweALT  7951  ttrclse  9680  tz9.12lem2  9741  hsmexlem4  10382  hsmexlem5  10383  zorn2lem7  10455  uniimadom  10497  zexALT  12549  psdmul  22053  fbasrn  23771  oldf  27765  madefi  27824  negsproplem2  27935  precsexlem10  28118  seqsex  28179  noseqex  28183  zsex  28268  dimval  33596  dimvalfi  33597  onvf1odlem4  35093  onvf1od  35094  fnwe2lem2  43040  relpfr  44944  orbitex  44945  permaxpow  44999  permaxun  45001  permac8prim  45004  setrec2fun  49681