Theorem imaexg 5090

Description: The image of a set is a set. Theorem 3.17 of [Monk1] p. 39. (Contributed by NM, 24-Jul-1995.)

Assertion

Ref	Expression
imaexg	|- ( A e. V -> ( A " B ) e. _V )

Proof of Theorem imaexg

Step	Hyp	Ref	Expression
1		imassrn 5087	. 2 |- ( A " B ) C_ ran A
2		rnexg 4997	. 2 |- ( A e. V -> ran A e. _V )
3		ssexg 4228	. 2 |- ( ( ( A " B ) C_ ran A /\ ran A e. _V ) -> ( A " B ) e. _V )
4	1, 2, 3	sylancr 414	1 |- ( A e. V -> ( A " B ) e. _V )

Syntax hints:   -> wi 4   e. wcel 2202   _Vcvv 2802   C_ wss 3200   ran crn 4726   "cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  imaex  5091  ecexg  6705  fopwdom  7021  isinfinf  7085  isunitd  14119