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Theorem fvssunirng 5404
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 2663 . . . . 5 𝑥 ∈ V
2 brelrng 4740 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
323exp 1165 . . . . 5 (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹)))
41, 3mpi 15 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹))
5 elssuni 3734 . . . 4 (𝑥 ∈ ran 𝐹𝑥 ran 𝐹)
64, 5syl6 33 . . 3 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ran 𝐹))
76alrimiv 1830 . 2 (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥𝑥 ran 𝐹))
8 fvss 5403 . 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 1314  wcel 1465  Vcvv 2660  wss 3041   cuni 3706   class class class wbr 3899  ran crn 4510  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-cnv 4517  df-dm 4519  df-rn 4520  df-iota 5058  df-fv 5101
This theorem is referenced by:  fvexg  5408  strfvssn  11908  xmetunirn  12454  mopnval  12538
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