Theorem funimaex 6613

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5232. 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 6612	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 703	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2145  Vcvv 3457   “ cima 5655  Fun wfun 6519

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-fun 6527

This theorem is referenced by:  isarep2  6615  isofr  7330  isose  7331  f1opw  7656  f1oweALT  7957  ttrclse  9684  tz9.12lem2  9748  hsmexlem4  10401  hsmexlem5  10402  zorn2lem7  10474  uniimadom  10516  zexALT  12602  psdmul  22289  fbasrn  24002  oldf  27988  madefi  28064  negsproplem2  28180  precsexlem10  28367  seqsex  28436  noseqex  28440  zsex  28531  dimval  33908  dimvalfi  33909  onvf1odlem4  35461  onvf1od  35462  fnwe2lem2  43640  relpfr  45528  orbitex  45529  permaxpow  45583  permaxun  45585  permac8prim  45588  setrec2fun  50321