Theorem funimaex 6652

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5236. 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 6651	. 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 2147  Vcvv 3454   “ cima 5659  Fun wfun 6555

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-pr 5396

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 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-fun 6563

This theorem is used by:  isarep2  6654  isofr  7376  isose  7377  f1opw  7704  f1oweALT  8011  fnwe2lem3  8177  ttrclse  9758  tz9.12lem2  9825  setrec2fun  10003  hsmexlem4  10537  hsmexlem5  10538  zorn2lem7  10610  uniimadom  10658  zexALT  12743  psdmul  22520  fbasrn  24236  oldf  28256  negsproplem2  28448  precsexlem10  28635  seqsex  28704  noseqex  28708  zsex  28799  dimval  34316  dimvalfi  34317  onvf1odlem4  36047  onvf1od  36048  relpfr  46150  orbitex  46151  permaxpow  46205  permaxun  46207  permac8prim  46210