Theorem fvssunirng 5532

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 2742	. . . . 5 |- x e. _V
2		brelrng 4860	. . . . . 6 |- ( ( A e. _V /\ x e. _V /\ A F x ) -> x e. ran F )
3	2	3exp 1202	. . . . 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 3839	. . . 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 1874	. 2 |- ( A e. _V -> A. x ( A F x -> x C_ U. ran F ) )
8		fvss 5531	. 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 1351   e. wcel 2148   _Vcvv 2739   C_ wss 3131   U.cuni 3811   class class class wbr 4005   ran crn 4629   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-cnv 4636  df-dm 4638  df-rn 4639  df-iota 5180  df-fv 5226

This theorem is referenced by:  fvexg  5536  strfvssn  12486  ptex  12718  xmetunirn  13943  mopnval  14027