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Theorem fvssunirng 5218
 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 2577 . . . . 5 𝑥 ∈ V
2 brelrng 4593 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
323exp 1114 . . . . 5 (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹)))
41, 3mpi 15 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹))
5 elssuni 3636 . . . 4 (𝑥 ∈ ran 𝐹𝑥 ran 𝐹)
64, 5syl6 33 . . 3 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ran 𝐹))
76alrimiv 1770 . 2 (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥𝑥 ran 𝐹))
8 fvss 5217 . 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 1257   ∈ wcel 1409  Vcvv 2574   ⊆ wss 2945  ∪ cuni 3608   class class class wbr 3792  ran crn 4374  ‘cfv 4930 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-cnv 4381  df-dm 4383  df-rn 4384  df-iota 4895  df-fv 4938 This theorem is referenced by:  fvexg  5222
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