Theorem funimaex 6443

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 5192. 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 6442	. 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ V) → (𝐴 “ 𝐵) ∈ V)
3	1, 2	mpan2 689	1 ⊢ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3496   “ cima 5560  Fun wfun 6351

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-fun 6359

This theorem is referenced by:  isarep2  6445  isofr  7097  isose  7098  f1opw  7403  f1oweALT  7675  tz9.12lem2  9219  hsmexlem4  9853  hsmexlem5  9854  zorn2lem7  9926  uniimadom  9968  zexALT  12004  fbasrn  22494  dimval  31003  dimvalfi  31004  fnwe2lem2  39658  setrec2fun  44802