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Theorem fconst5 5951
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 5097 . . . 4  |-  ( F  =  ( A  X.  { B } )  ->  ran  F  =  ran  ( A  X.  { B }
) )
2 rnxp 5301 . . . . 5  |-  ( A  =/=  (/)  ->  ran  ( A  X.  { B }
)  =  { B } )
32eqeq2d 2449 . . . 4  |-  ( A  =/=  (/)  ->  ( ran  F  =  ran  ( A  X.  { B }
)  <->  ran  F  =  { B } ) )
41, 3syl5ib 212 . . 3  |-  ( A  =/=  (/)  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
54adantl 454 . 2  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
6 df-fo 5462 . . . . . . 7  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
7 fof 5655 . . . . . . 7  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
86, 7sylbir 206 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
9 fconst2g 5948 . . . . . 6  |-  ( B  e.  _V  ->  ( F : A --> { B } 
<->  F  =  ( A  X.  { B }
) ) )
108, 9syl5ib 212 . . . . 5  |-  ( B  e.  _V  ->  (
( F  Fn  A  /\  ran  F  =  { B } )  ->  F  =  ( A  X.  { B } ) ) )
1110exp3a 427 . . . 4  |-  ( B  e.  _V  ->  ( F  Fn  A  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
1211adantrd 456 . . 3  |-  ( B  e.  _V  ->  (
( F  Fn  A  /\  A  =/=  (/) )  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
13 fnrel 5545 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
14 snprc 3873 . . . . . 6  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
15 relrn0 5130 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
1615biimprd 216 . . . . . . . . 9  |-  ( Rel 
F  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
1716adantl 454 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
18 eqeq2 2447 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
1918adantr 453 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
20 xpeq2 4895 . . . . . . . . . . 11  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  ( A  X.  (/) ) )
21 xp0 5293 . . . . . . . . . . 11  |-  ( A  X.  (/) )  =  (/)
2220, 21syl6eq 2486 . . . . . . . . . 10  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  (/) )
2322eqeq2d 2449 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( F  =  ( A  X.  { B } )  <->  F  =  (/) ) )
2423adantr 453 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( F  =  ( A  X.  { B } )  <->  F  =  (/) ) )
2517, 19, 243imtr4d 261 . . . . . . 7  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) )
2625ex 425 . . . . . 6  |-  ( { B }  =  (/)  ->  ( Rel  F  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2714, 26sylbi 189 . . . . 5  |-  ( -.  B  e.  _V  ->  ( Rel  F  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2813, 27syl5 31 . . . 4  |-  ( -.  B  e.  _V  ->  ( F  Fn  A  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2928adantrd 456 . . 3  |-  ( -.  B  e.  _V  ->  ( ( F  Fn  A  /\  A  =/=  (/) )  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
3012, 29pm2.61i 159 . 2  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) )
315, 30impbid 185 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 178    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   (/)c0 3630   {csn 3816    X. cxp 4878   ran crn 4881   Rel wrel 4885    Fn wfn 5451   -->wf 5452   -onto->wfo 5454
This theorem is referenced by:  nvo00  22264  esumnul  24445  esum0  24446  volsupnfl  26253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464
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