Theorem fvssunirng 5569

Description: The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)

Assertion

Ref	Expression
fvssunirng	|- ( A e. _V -> ( F ` A ) C_ U. ran F )

Proof of Theorem fvssunirng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2763	. . . . 5 |- x e. _V
2		brelrng 4893	. . . . . 6 |- ( ( A e. _V /\ x e. _V /\ A F x ) -> x e. ran F )
3	2	3exp 1204	. . . . 5 |- ( A e. _V -> ( x e. _V -> ( A F x -> x e. ran F ) ) )
4	1, 3	mpi 15	. . . 4 |- ( A e. _V -> ( A F x -> x e. ran F ) )
5		elssuni 3863	. . . 4 |- ( x e. ran F -> x C_ U. ran F )
6	4, 5	syl6 33	. . 3 |- ( A e. _V -> ( A F x -> x C_ U. ran F ) )
7	6	alrimiv 1885	. 2 |- ( A e. _V -> A. x ( A F x -> x C_ U. ran F ) )
8		fvss 5568	. 2 |- ( A. x ( A F x -> x C_ U. ran F ) -> ( F ` A ) C_ U. ran F )
9	7, 8	syl 14	1 |- ( A e. _V -> ( F ` A ) C_ U. ran F )

Syntax hints:   -> wi 4   A.wal 1362   e. wcel 2164   _Vcvv 2760   C_ wss 3153   U.cuni 3835   class class class wbr 4029   ran crn 4660   ` cfv 5254

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-cnv 4667  df-dm 4669  df-rn 4670  df-iota 5215  df-fv 5262

This theorem is referenced by:  fvexg  5573  strfvssn  12640  ptex  12875  xmetunirn  14526  mopnval  14610