Theorem funimaex 6624

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5236. 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 6623	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 704	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Vcvv 3453   “ cima 5662  Fun wfun 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6539

This theorem is used by:  isarep2  6626  isofr  7347  isose  7348  f1opw  7674  f1oweALT  7973  ttrclse  9710  tz9.12lem2  9774  hsmexlem4  10435  hsmexlem5  10436  zorn2lem7  10508  uniimadom  10556  zexALT  12639  psdmul  22400  fbasrn  24116  oldf  28110  negsproplem2  28302  precsexlem10  28489  seqsex  28558  noseqex  28562  zsex  28653  dimval  34119  dimvalfi  34120  onvf1odlem4  35711  onvf1od  35712  fnwe2lem2  43900  relpfr  45785  orbitex  45786  permaxpow  45840  permaxun  45842  permac8prim  45845  setrec2fun  50626