Theorem funimaex 6588

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5226. 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 6587	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 692	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3442   “ cima 5635  Fun wfun 6494

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-fun 6502

This theorem is referenced by:  isarep2  6590  isofr  7298  isose  7299  f1opw  7624  f1oweALT  7926  ttrclse  9648  tz9.12lem2  9712  hsmexlem4  10351  hsmexlem5  10352  zorn2lem7  10424  uniimadom  10466  zexALT  12520  psdmul  22121  fbasrn  23840  oldf  27845  madefi  27921  negsproplem2  28037  precsexlem10  28224  seqsex  28293  noseqex  28297  zsex  28388  dimval  33777  dimvalfi  33778  onvf1odlem4  35319  onvf1od  35320  fnwe2lem2  43402  relpfr  45304  orbitex  45305  permaxpow  45359  permaxun  45361  permac8prim  45364  setrec2fun  50045