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Theorem fvssunirng 5133
Description: The result of a function value is always a subset of the union of the range, if the input is a set. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 24-May-2019.)
Assertion
Ref Expression
fvssunirng (A V → (𝐹A) ⊆ ran 𝐹)

Proof of Theorem fvssunirng
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . 5 x V
2 brelrng 4508 . . . . . 6 ((A V x V A𝐹x) → x ran 𝐹)
323exp 1102 . . . . 5 (A V → (x V → (A𝐹xx ran 𝐹)))
41, 3mpi 15 . . . 4 (A V → (A𝐹xx ran 𝐹))
5 elssuni 3599 . . . 4 (x ran 𝐹x ran 𝐹)
64, 5syl6 29 . . 3 (A V → (A𝐹xx ran 𝐹))
76alrimiv 1751 . 2 (A V → x(A𝐹xx ran 𝐹))
8 fvss 5132 . 2 (x(A𝐹xx ran 𝐹) → (𝐹A) ⊆ ran 𝐹)
97, 8syl 14 1 (A V → (𝐹A) ⊆ ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1240   wcel 1390  Vcvv 2551  wss 2911   cuni 3571   class class class wbr 3755  ran crn 4289  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-cnv 4296  df-dm 4298  df-rn 4299  df-iota 4810  df-fv 4853
This theorem is referenced by:  fvexg  5137
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