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Theorem relfvssunirn 5116
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 → (𝐹A) ⊆ ran 𝐹)

Proof of Theorem relfvssunirn
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 relelrn 4497 . . . . 5 ((Rel 𝐹 A𝐹x) → x ran 𝐹)
21ex 108 . . . 4 (Rel 𝐹 → (A𝐹xx ran 𝐹))
3 elssuni 3582 . . . 4 (x ran 𝐹x ran 𝐹)
42, 3syl6 29 . . 3 (Rel 𝐹 → (A𝐹xx ran 𝐹))
54alrimiv 1736 . 2 (Rel 𝐹x(A𝐹xx ran 𝐹))
6 fvss 5114 . 2 (x(A𝐹xx ran 𝐹) → (𝐹A) ⊆ ran 𝐹)
75, 6syl 14 1 (Rel 𝐹 → (𝐹A) ⊆ ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226   wcel 1374  wss 2894   cuni 3554   class class class wbr 3738  ran crn 4273  Rel wrel 4277  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-dm 4282  df-rn 4283  df-iota 4794  df-fv 4837
This theorem is referenced by:  relrnfvex  5118
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