Theorem relfvssunirn 5191

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 4570	. . . . 5 |- ( ( Rel F /\ A F x ) -> x e. ran F )
2	1	ex 108	. . . 4 |- ( Rel F -> ( A F x -> x e. ran F ) )
3		elssuni 3608	. . . 4 |- ( x e. ran F -> x C_ U. ran F )
4	2, 3	syl6 29	. . 3 |- ( Rel F -> ( A F x -> x C_ U. ran F ) )
5	4	alrimiv 1754	. 2 |- ( Rel F -> A. x ( A F x -> x C_ U. ran F ) )
6		fvss 5189	. 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 1241   e. wcel 1393   C_ wss 2917   U.cuni 3580   class class class wbr 3764   ran crn 4346   Rel wrel 4350   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356  df-iota 4867  df-fv 4910

This theorem is referenced by:  relrnfvex  5193