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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

Proof of Theorem relfvssunirn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 relelrn 4570 . . . . 5
21ex 108 . . . 4
3 elssuni 3608 . . . 4
42, 3syl6 29 . . 3
54alrimiv 1754 . 2
6 fvss 5189 . 2
75, 6syl 14 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1241   wcel 1393   wss 2917  cuni 3580   class class class wbr 3764   crn 4346   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
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