MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funimaex Structured version   Visualization version   GIF version

Theorem funimaex 5876
Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 4693. 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 5875 . 2 ((Fun 𝐴𝐵 ∈ V) → (𝐴𝐵) ∈ V)
31, 2mpan2 702 1 (Fun 𝐴 → (𝐴𝐵) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  Vcvv 3172  cima 5031  Fun wfun 5784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pr 4828
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-fun 5792
This theorem is referenced by:  isarep2  5878  isofr  6470  isose  6471  f1opw  6764  f1oweALT  7020  tz9.12lem2  8511  hsmexlem4  9111  hsmexlem5  9112  zorn2lem7  9184  uniimadom  9222  zexALT  11229  fbasrn  21440  fnwe2lem2  36442
  Copyright terms: Public domain W3C validator