Theorem funimaex 6569

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5217. 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 6568	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 691	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2111  Vcvv 3436   “ cima 5619  Fun wfun 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-fun 6483

This theorem is referenced by:  isarep2  6571  isofr  7276  isose  7277  f1opw  7602  f1oweALT  7904  ttrclse  9617  tz9.12lem2  9681  hsmexlem4  10320  hsmexlem5  10321  zorn2lem7  10393  uniimadom  10435  zexALT  12488  psdmul  22082  fbasrn  23800  oldf  27799  madefi  27859  negsproplem2  27972  precsexlem10  28155  seqsex  28216  noseqex  28220  zsex  28305  dimval  33611  dimvalfi  33612  onvf1odlem4  35148  onvf1od  35149  fnwe2lem2  43090  relpfr  44993  orbitex  44994  permaxpow  45048  permaxun  45050  permac8prim  45053  setrec2fun  49730