Theorem relfvssunirn 5615

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 4933	. . . . 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 3892	. . . 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 1898	. 2 |- ( Rel F -> A. x ( A F x -> x C_ U. ran F ) )
6		fvss 5613	. 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 2178   C_ wss 3174   U.cuni 3864   class class class wbr 4059   ran crn 4694   Rel wrel 4698   ` cfv 5290

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-cnv 4701  df-dm 4703  df-rn 4704  df-iota 5251  df-fv 5298

This theorem is referenced by:  relrnfvex  5617