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Theorem fconst5 6109
Description: Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.)
Assertion
Ref Expression
fconst5  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B } )  <->  ran  F  =  { B } ) )

Proof of Theorem fconst5
StepHypRef Expression
1 rneq 5049 . . . 4  |-  ( F  =  ( A  X.  { B } )  ->  ran  F  =  ran  ( A  X.  { B }
) )
2 rnxp 5255 . . . . 5  |-  ( A  =/=  (/)  ->  ran  ( A  X.  { B }
)  =  { B } )
32eqeq2d 2416 . . . 4  |-  ( A  =/=  (/)  ->  ( ran  F  =  ran  ( A  X.  { B }
)  <->  ran  F  =  { B } ) )
41, 3syl5ib 219 . . 3  |-  ( A  =/=  (/)  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
54adantl 464 . 2  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
6 df-fo 5575 . . . . . . 7  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
7 fof 5778 . . . . . . 7  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
86, 7sylbir 213 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
9 fconst2g 6106 . . . . . 6  |-  ( B  e.  _V  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
108, 9syl5ib 219 . . . . 5  |-  ( B  e.  _V  ->  (
( F  Fn  A  /\  ran  F  =  { B } )  ->  F  =  ( A  X.  { B } ) ) )
1110expd 434 . . . 4  |-  ( B  e.  _V  ->  ( F  Fn  A  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
1211adantrd 466 . . 3  |-  ( B  e.  _V  ->  (
( F  Fn  A  /\  A  =/=  (/) )  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
13 fnrel 5660 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
14 snprc 4035 . . . . . 6  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
15 relrn0 5081 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
1615biimprd 223 . . . . . . . . 9  |-  ( Rel 
F  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
1716adantl 464 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
18 eqeq2 2417 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
1918adantr 463 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
20 xpeq2 4838 . . . . . . . . . . 11  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  ( A  X.  (/) ) )
21 xp0 5243 . . . . . . . . . . 11  |-  ( A  X.  (/) )  =  (/)
2220, 21syl6eq 2459 . . . . . . . . . 10  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  (/) )
2322eqeq2d 2416 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( F  =  ( A  X.  { B } )  <->  F  =  (/) ) )
2423adantr 463 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( F  =  ( A  X.  { B } )  <->  F  =  (/) ) )
2517, 19, 243imtr4d 268 . . . . . . 7  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) )
2625ex 432 . . . . . 6  |-  ( { B }  =  (/)  ->  ( Rel  F  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2714, 26sylbi 195 . . . . 5  |-  ( -.  B  e.  _V  ->  ( Rel  F  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2813, 27syl5 30 . . . 4  |-  ( -.  B  e.  _V  ->  ( F  Fn  A  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2928adantrd 466 . . 3  |-  ( -.  B  e.  _V  ->  ( ( F  Fn  A  /\  A  =/=  (/) )  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
3012, 29pm2.61i 164 . 2  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) )
315, 30impbid 190 1  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B } )  <->  ran  F  =  { B } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3059   (/)c0 3738   {csn 3972    X. cxp 4821   ran crn 4824   Rel wrel 4828    Fn wfn 5564   -->wf 5565   -onto->wfo 5567
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-fo 5575  df-fv 5577
This theorem is referenced by:  nvo00  26090  esumnul  28495  esum0  28496  volsupnfl  31431  rnmptc  36824
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