Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  fvssunirng Structured version   GIF version

Theorem fvssunirng 5111
 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 2534 . . . . 5 x V
2 brelrng 4488 . . . . . 6 ((A V x V A𝐹x) → x ran 𝐹)
323exp 1087 . . . . 5 (A V → (x V → (A𝐹xx ran 𝐹)))
41, 3mpi 15 . . . 4 (A V → (A𝐹xx ran 𝐹))
5 elssuni 3578 . . . 4 (x ran 𝐹x ran 𝐹)
64, 5syl6 29 . . 3 (A V → (A𝐹xx ran 𝐹))
76alrimiv 1732 . 2 (A V → x(A𝐹xx ran 𝐹))
8 fvss 5110 . 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 1224   ∈ wcel 1370  Vcvv 2531   ⊆ wss 2890  ∪ cuni 3550   class class class wbr 3734  ran crn 4269  ‘cfv 4825 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-cnv 4276  df-dm 4278  df-rn 4279  df-iota 4790  df-fv 4833 This theorem is referenced by:  fvexg  5115
 Copyright terms: Public domain W3C validator