Theorem funimaex 6625

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5239. 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 6624	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 703	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2143  Vcvv 3455   “ cima 5666  Fun wfun 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-fun 6540

This theorem is referenced by:  isarep2  6627  isofr  7342  isose  7343  f1opw  7668  f1oweALT  7970  ttrclse  9697  tz9.12lem2  9761  hsmexlem4  10414  hsmexlem5  10415  zorn2lem7  10487  uniimadom  10529  zexALT  12612  psdmul  22310  fbasrn  24022  oldf  28011  madefi  28087  negsproplem2  28203  precsexlem10  28390  seqsex  28459  noseqex  28463  zsex  28554  dimval  33972  dimvalfi  33973  onvf1odlem4  35571  onvf1od  35572  fnwe2lem2  43761  relpfr  45646  orbitex  45647  permaxpow  45701  permaxun  45703  permac8prim  45706  setrec2fun  50453