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Theorem fvssunirng 5429
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 2684 . . . . 5 𝑥 ∈ V
2 brelrng 4765 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
323exp 1180 . . . . 5 (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹)))
41, 3mpi 15 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹))
5 elssuni 3759 . . . 4 (𝑥 ∈ ran 𝐹𝑥 ran 𝐹)
64, 5syl6 33 . . 3 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ran 𝐹))
76alrimiv 1846 . 2 (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥𝑥 ran 𝐹))
8 fvss 5428 . 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 1329  wcel 1480  Vcvv 2681  wss 3066   cuni 3731   class class class wbr 3924  ran crn 4535  cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-cnv 4542  df-dm 4544  df-rn 4545  df-iota 5083  df-fv 5126
This theorem is referenced by:  fvexg  5433  strfvssn  11970  xmetunirn  12516  mopnval  12600
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