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Theorem funimaex 5256
Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement. 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 → (𝐴𝐵) ∈ V)

Proof of Theorem funimaex
StepHypRef Expression
1 zfrep5.1 . 2 𝐵 ∈ V
2 funimaexg 5255 . 2 ((Fun 𝐴𝐵 ∈ V) → (𝐴𝐵) ∈ V)
31, 2mpan2 422 1 (Fun 𝐴 → (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2128  Vcvv 2712  cima 4590  Fun wfun 5165
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-pow 4136  ax-pr 4170
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-fun 5173
This theorem is referenced by:  isarep2  5258  isose  5772  f1opw  6028
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