Theorem funimaex 6619

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 6618	. 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 3451   “ cima 5654  Fun wfun 6525

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 2733  ax-rep 5232  ax-sep 5249  ax-pr 5391

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 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-fun 6533

This theorem is used by:  isarep2  6621  isofr  7342  isose  7343  f1opw  7669  f1oweALT  7973  ttrclse  9712  tz9.12lem2  9778  setrec2fun  9954  hsmexlem4  10488  hsmexlem5  10489  zorn2lem7  10561  uniimadom  10609  zexALT  12694  psdmul  22467  fbasrn  24183  oldf  28205  negsproplem2  28397  precsexlem10  28584  seqsex  28653  noseqex  28657  zsex  28748  dimval  34215  dimvalfi  34216  onvf1odlem4  35858  onvf1od  35859  fnwe2lem2  44011  relpfr  45896  orbitex  45897  permaxpow  45951  permaxun  45953  permac8prim  45956