Theorem fvssunirng 5436

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 2689	. . . . 5 |- x e. _V
2		brelrng 4770	. . . . . 6 |- ( ( A e. _V /\ x e. _V /\ A F x ) -> x e. ran F )
3	2	3exp 1180	. . . . 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 3764	. . . 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 1846	. 2 |- ( A e. _V -> A. x ( A F x -> x C_ U. ran F ) )
8		fvss 5435	. 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 1329   e. wcel 1480   _Vcvv 2686   C_ wss 3071   U.cuni 3736   class class class wbr 3929   ran crn 4540   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-cnv 4547  df-dm 4549  df-rn 4550  df-iota 5088  df-fv 5131

This theorem is referenced by:  fvexg  5440  strfvssn  11981  xmetunirn  12527  mopnval  12611