Theorem relfvssunirn 5594

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

Assertion

Ref	Expression
relfvssunirn	|- ( Rel F -> ( F ` A ) C_ U. ran F )

Proof of Theorem relfvssunirn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relelrn 4915	. . . . 5 |- ( ( Rel F /\ A F x ) -> x e. ran F )
2	1	ex 115	. . . 4 |- ( Rel F -> ( A F x -> x e. ran F ) )
3		elssuni 3878	. . . 4 |- ( x e. ran F -> x C_ U. ran F )
4	2, 3	syl6 33	. . 3 |- ( Rel F -> ( A F x -> x C_ U. ran F ) )
5	4	alrimiv 1897	. 2 |- ( Rel F -> A. x ( A F x -> x C_ U. ran F ) )
6		fvss 5592	. 2 |- ( A. x ( A F x -> x C_ U. ran F ) -> ( F ` A ) C_ U. ran F )
7	5, 6	syl 14	1 |- ( Rel F -> ( F ` A ) C_ U. ran F )

Syntax hints:   -> wi 4   A.wal 1371   e. wcel 2176   C_ wss 3166   U.cuni 3850   class class class wbr 4045   ran crn 4677   Rel wrel 4681   ` cfv 5272

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-cnv 4684  df-dm 4686  df-rn 4687  df-iota 5233  df-fv 5280

This theorem is referenced by:  relrnfvex  5596