Theorem funimaex 3572

Description: The image of a set under any function is also a set. Equivalent of Axiom of Replacement ax-rep 2689. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29.

Hypothesis

Ref	Expression
zfrep5.1	|- B e. V

Assertion

Ref	Expression
funimaex	|- (Fun A -> (A"B) e. V)

Proof of Theorem funimaex

Step	Hyp	Ref	Expression
1		zfrep5.1	. 2 |- B e. V
2		funimaexg 3571	. 2 |- ((Fun A /\ B e. V) -> (A"B) e. V)
3	1, 2	mpan2 695	1 |- (Fun A -> (A"B) e. V)

Syntax hints:   -> wi 3   e. wcel 957  Vcvv 1808  "cima 3169  Fun wfun 3172

This theorem is referenced by:  isarep2 3574  isofrlem 3896  f1oweALT 3901  tz7.44-3 3925  tz9.12lem2 4643  zorn2lem7 4777  uniimadom 4793

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188

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