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Theorem funimaex 6209
Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4994. 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 6208 . 2 ((Fun 𝐴𝐵 ∈ V) → (𝐴𝐵) ∈ V)
31, 2mpan2 684 1 (Fun 𝐴 → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  Vcvv 3414  cima 5345  Fun wfun 6117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-fun 6125
This theorem is referenced by:  isarep2  6211  isofr  6847  isose  6848  f1opw  7149  f1oweALT  7412  tz9.12lem2  8928  hsmexlem4  9566  hsmexlem5  9567  zorn2lem7  9639  uniimadom  9681  zexALT  11723  fbasrn  22058  fnwe2lem2  38464  setrec2fun  43334
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