Theorem funimaex 6605

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5226. 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 6604	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 701	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2141  Vcvv 3453   “ cima 5648  Fun wfun 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-fun 6519

This theorem is referenced by:  isarep2  6607  isofr  7322  isose  7323  f1opw  7648  f1oweALT  7949  ttrclse  9679  tz9.12lem2  9743  hsmexlem4  10383  hsmexlem5  10384  zorn2lem7  10456  uniimadom  10498  zexALT  12585  psdmul  22211  fbasrn  23924  oldf  27907  madefi  27983  negsproplem2  28099  precsexlem10  28286  seqsex  28355  noseqex  28359  zsex  28450  dimval  33859  dimvalfi  33860  onvf1odlem4  35413  onvf1od  35414  fnwe2lem2  43592  relpfr  45494  orbitex  45495  permaxpow  45549  permaxun  45551  permac8prim  45554  setrec2fun  50277