Theorem funimaex 6586

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5212. 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 6585	. 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 3429   “ cima 5634  Fun wfun 6492

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 2708  ax-rep 5212  ax-sep 5231  ax-pr 5375

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 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500

This theorem is referenced by:  isarep2  6588  isofr  7297  isose  7298  f1opw  7623  f1oweALT  7925  ttrclse  9648  tz9.12lem2  9712  hsmexlem4  10351  hsmexlem5  10352  zorn2lem7  10424  uniimadom  10466  zexALT  12544  psdmul  22132  fbasrn  23849  oldf  27829  madefi  27905  negsproplem2  28021  precsexlem10  28208  seqsex  28277  noseqex  28281  zsex  28372  dimval  33745  dimvalfi  33746  onvf1odlem4  35288  onvf1od  35289  fnwe2lem2  43479  relpfr  45381  orbitex  45382  permaxpow  45436  permaxun  45438  permac8prim  45441  setrec2fun  50167