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Theorem fconst5 5731
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 4904 . . . 4  |-  ( F  =  ( A  X.  { B } )  ->  ran  F  =  ran  ( A  X.  { B }
) )
2 rnxp 5106 . . . . 5  |-  ( A  =/=  (/)  ->  ran  ( A  X.  { B }
)  =  { B } )
32eqeq2d 2294 . . . 4  |-  ( A  =/=  (/)  ->  ( ran  F  =  ran  ( A  X.  { B }
)  <->  ran  F  =  { B } ) )
41, 3syl5ib 210 . . 3  |-  ( A  =/=  (/)  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
54adantl 452 . 2  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
6 df-fo 5261 . . . . . . 7  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
7 fof 5451 . . . . . . 7  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
86, 7sylbir 204 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
9 fconst2g 5728 . . . . . 6  |-  ( B  e.  _V  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
108, 9syl5ib 210 . . . . 5  |-  ( B  e.  _V  ->  (
( F  Fn  A  /\  ran  F  =  { B } )  ->  F  =  ( A  X.  { B } ) ) )
1110exp3a 425 . . . 4  |-  ( B  e.  _V  ->  ( F  Fn  A  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
1211adantrd 454 . . 3  |-  ( B  e.  _V  ->  (
( F  Fn  A  /\  A  =/=  (/) )  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
13 fnrel 5342 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
14 snprc 3695 . . . . . 6  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
15 relrn0 4937 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
1615biimprd 214 . . . . . . . . 9  |-  ( Rel 
F  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
1716adantl 452 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
18 eqeq2 2292 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
1918adantr 451 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
20 xpeq2 4704 . . . . . . . . . . 11  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  ( A  X.  (/) ) )
21 xp0 5098 . . . . . . . . . . 11  |-  ( A  X.  (/) )  =  (/)
2220, 21syl6eq 2331 . . . . . . . . . 10  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  (/) )
2322eqeq2d 2294 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( F  =  ( A  X.  { B } )  <->  F  =  (/) ) )
2423adantr 451 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( F  =  ( A  X.  { B } )  <->  F  =  (/) ) )
2517, 19, 243imtr4d 259 . . . . . . 7  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) )
2625ex 423 . . . . . 6  |-  ( { B }  =  (/)  ->  ( Rel  F  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2714, 26sylbi 187 . . . . 5  |-  ( -.  B  e.  _V  ->  ( Rel  F  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2813, 27syl5 28 . . . 4  |-  ( -.  B  e.  _V  ->  ( F  Fn  A  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2928adantrd 454 . . 3  |-  ( -.  B  e.  _V  ->  ( ( F  Fn  A  /\  A  =/=  (/) )  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
3012, 29pm2.61i 156 . 2  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) )
315, 30impbid 183 1  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B } )  <->  ran  F  =  { B } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   (/)c0 3455   {csn 3640    X. cxp 4687   ran crn 4690   Rel wrel 4694    Fn wfn 5250   -->wf 5251   -onto->wfo 5253
This theorem is referenced by:  nvo00  21339  esumnul  23427  esum0  23428
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
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