Theorem fvssunirng 5705

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 2824	. . . . 5 |- x e. _V
2		brelrng 5008	. . . . . 6 |- ( ( A e. _V /\ x e. _V /\ A F x ) -> x e. ran F )
3	2	3exp 1233	. . . . 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 3958	. . . 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 1927	. 2 |- ( A e. _V -> A. x ( A F x -> x C_ U. ran F ) )
8		fvss 5704	. 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 1400   e. wcel 2209   _Vcvv 2821   C_ wss 3220   U.cuni 3930   class class class wbr 4125   ran crn 4770   ` cfv 5372

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-cnv 4777  df-dm 4779  df-rn 4780  df-iota 5332  df-fv 5380

This theorem is referenced by:  fvexg  5709  ovssunirng  6110  strfvssn  13352  ptex  13595  prdsvallem  13598  prdsval  14150  xmetunirn  15382  mopnval  15466