Theorem funimaex 6627

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5231. 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 6626	. 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 2144  Vcvv 3450   “ cima 5653  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6540

This theorem is used by:  isarep2  6629  isofr  7350  isose  7351  f1opw  7677  f1oweALT  7984  fnwe2lem3  8149  ttrclse  9731  tz9.12lem2  9798  setrec2fun  9976  hsmexlem4  10510  hsmexlem5  10511  zorn2lem7  10583  uniimadom  10631  zexALT  12716  psdmul  22492  fbasrn  24208  oldf  28228  negsproplem2  28420  precsexlem10  28607  seqsex  28676  noseqex  28680  zsex  28771  dimval  34279  dimvalfi  34280  onvf1odlem4  35953  onvf1od  35954  relpfr  46037  orbitex  46038  permaxpow  46092  permaxun  46094  permac8prim  46097