Theorem funimaex 6580

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5206. 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 6579	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 697	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2119  Vcvv 3432   “ cima 5628  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-fun 6494

This theorem is referenced by:  isarep2  6582  isofr  7293  isose  7294  f1opw  7619  f1oweALT  7921  ttrclse  9646  tz9.12lem2  9710  hsmexlem4  10349  hsmexlem5  10350  zorn2lem7  10422  uniimadom  10464  zexALT  12542  psdmul  22161  fbasrn  23874  oldf  27854  madefi  27930  negsproplem2  28046  precsexlem10  28233  seqsex  28302  noseqex  28306  zsex  28397  dimval  33792  dimvalfi  33793  onvf1odlem4  35341  onvf1od  35342  fnwe2lem2  43503  relpfr  45405  orbitex  45406  permaxpow  45460  permaxun  45462  permac8prim  45465  setrec2fun  50189