Theorem relfvssunirn 5512

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 4847	. . . . 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 3824	. . . 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 1867	. 2 |- ( Rel F -> A. x ( A F x -> x C_ U. ran F ) )
6		fvss 5510	. 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 1346   e. wcel 2141   C_ wss 3121   U.cuni 3796   class class class wbr 3989   ran crn 4612   Rel wrel 4616   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-rn 4622  df-iota 5160  df-fv 5206

This theorem is referenced by:  relrnfvex  5514