Theorem relfvssunirn 5486

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 4824	. . . . 5 |- ( ( Rel F /\ A F x ) -> x e. ran F )
2	1	ex 114	. . . 4 |- ( Rel F -> ( A F x -> x e. ran F ) )
3		elssuni 3802	. . . 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 1854	. 2 |- ( Rel F -> A. x ( A F x -> x C_ U. ran F ) )
6		fvss 5484	. 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 1333   e. wcel 2128   C_ wss 3102   U.cuni 3774   class class class wbr 3967   ran crn 4589   Rel wrel 4593   ` cfv 5172

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-rn 4599  df-iota 5137  df-fv 5180

This theorem is referenced by:  relrnfvex  5488