Theorem funimaex 6580

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5224. 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 6579	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 691	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3440   “ cima 5627  Fun wfun 6486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494

This theorem is referenced by:  isarep2  6582  isofr  7288  isose  7289  f1opw  7614  f1oweALT  7916  ttrclse  9636  tz9.12lem2  9700  hsmexlem4  10339  hsmexlem5  10340  zorn2lem7  10412  uniimadom  10454  zexALT  12508  psdmul  22109  fbasrn  23828  oldf  27833  madefi  27909  negsproplem2  28025  precsexlem10  28212  seqsex  28281  noseqex  28285  zsex  28376  dimval  33757  dimvalfi  33758  onvf1odlem4  35300  onvf1od  35301  fnwe2lem2  43289  relpfr  45191  orbitex  45192  permaxpow  45246  permaxun  45248  permac8prim  45251  setrec2fun  49933