Theorem funimaex 6630

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5243. 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 6629	. 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 2146  Vcvv 3458   “ cima 5669  Fun wfun 6537

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 2148  ax-9 2156  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-pr 5409

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 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-fun 6545

This theorem is used by:  isarep2  6632  isofr  7351  isose  7352  f1opw  7679  f1oweALT  7978  ttrclse  9706  tz9.12lem2  9770  hsmexlem4  10431  hsmexlem5  10432  zorn2lem7  10504  uniimadom  10546  zexALT  12629  psdmul  22366  fbasrn  24078  oldf  28067  madefi  28143  negsproplem2  28259  precsexlem10  28446  seqsex  28515  noseqex  28519  zsex  28610  dimval  34022  dimvalfi  34023  onvf1odlem4  35614  onvf1od  35615  fnwe2lem2  43819  relpfr  45704  orbitex  45705  permaxpow  45759  permaxun  45761  permac8prim  45764  setrec2fun  50511