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Theorem fvssunirng 5190
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 2560 . . . . 5 𝑥 ∈ V
2 brelrng 4565 . . . . . 6 ((𝐴 ∈ V ∧ 𝑥 ∈ V ∧ 𝐴𝐹𝑥) → 𝑥 ∈ ran 𝐹)
323exp 1103 . . . . 5 (𝐴 ∈ V → (𝑥 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹)))
41, 3mpi 15 . . . 4 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ∈ ran 𝐹))
5 elssuni 3608 . . . 4 (𝑥 ∈ ran 𝐹𝑥 ran 𝐹)
64, 5syl6 29 . . 3 (𝐴 ∈ V → (𝐴𝐹𝑥𝑥 ran 𝐹))
76alrimiv 1754 . 2 (𝐴 ∈ V → ∀𝑥(𝐴𝐹𝑥𝑥 ran 𝐹))
8 fvss 5189 . 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 1241  wcel 1393  Vcvv 2557  wss 2917   cuni 3580   class class class wbr 3764  ran crn 4346  cfv 4902
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-cnv 4353  df-dm 4355  df-rn 4356  df-iota 4867  df-fv 4910
This theorem is referenced by:  fvexg  5194
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