Theorem funimaex 5378

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 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 5377	. 2 |- ( ( Fun A /\ B e. _V ) -> ( A " B ) e. _V )
3	1, 2	mpan2 425	1 |- ( Fun A -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2178   _Vcvv 2776   "cima 4696   Fun wfun 5284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-fun 5292

This theorem is referenced by:  isarep2  5380  isose  5913  f1opw  6176