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Theorem relfvssunirn 5134
 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 4513 . . . . 5 ((Rel 𝐹 A𝐹x) → x ran 𝐹)
21ex 108 . . . 4 (Rel 𝐹 → (A𝐹xx ran 𝐹))
3 elssuni 3599 . . . 4 (x ran 𝐹x ran 𝐹)
42, 3syl6 29 . . 3 (Rel 𝐹 → (A𝐹xx ran 𝐹))
54alrimiv 1751 . 2 (Rel 𝐹x(A𝐹xx ran 𝐹))
6 fvss 5132 . 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 1240   ∈ wcel 1390   ⊆ wss 2911  ∪ cuni 3571   class class class wbr 3755  ran crn 4289  Rel wrel 4293  ‘cfv 4845 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-iota 4810  df-fv 4853 This theorem is referenced by:  relrnfvex  5136
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