Theorem funimaex 6574

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5219. 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 2113  Vcvv 3437   “ cima 5622  Fun wfun 6480

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 2115  ax-9 2123  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6488

This theorem is referenced by:  isarep2  6576  isofr  7282  isose  7283  f1opw  7608  f1oweALT  7910  ttrclse  9624  tz9.12lem2  9688  hsmexlem4  10327  hsmexlem5  10328  zorn2lem7  10400  uniimadom  10442  zexALT  12495  psdmul  22082  fbasrn  23800  oldf  27799  madefi  27859  negsproplem2  27972  precsexlem10  28155  seqsex  28216  noseqex  28220  zsex  28305  dimval  33634  dimvalfi  33635  onvf1odlem4  35171  onvf1od  35172  fnwe2lem2  43169  relpfr  45072  orbitex  45073  permaxpow  45127  permaxun  45129  permac8prim  45132  setrec2fun  49818