Theorem funimaex 6570

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5218. 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 6569	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 691	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2109  Vcvv 3436   “ cima 5622  Fun wfun 6476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-fun 6484

This theorem is referenced by:  isarep2  6572  isofr  7279  isose  7280  f1opw  7605  f1oweALT  7907  ttrclse  9623  tz9.12lem2  9684  hsmexlem4  10323  hsmexlem5  10324  zorn2lem7  10396  uniimadom  10438  zexALT  12491  psdmul  22051  fbasrn  23769  oldf  27767  madefi  27827  negsproplem2  27940  precsexlem10  28123  seqsex  28184  noseqex  28188  zsex  28273  dimval  33567  dimvalfi  33568  onvf1odlem4  35083  onvf1od  35084  fnwe2lem2  43028  relpfr  44932  orbitex  44933  permaxpow  44987  permaxun  44989  permac8prim  44992  setrec2fun  49681