MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconst5 Structured version   Unicode version

Theorem fconst5 6116
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 5226 . . . 4  |-  ( F  =  ( A  X.  { B } )  ->  ran  F  =  ran  ( A  X.  { B }
) )
2 rnxp 5435 . . . . 5  |-  ( A  =/=  (/)  ->  ran  ( A  X.  { B }
)  =  { B } )
32eqeq2d 2481 . . . 4  |-  ( A  =/=  (/)  ->  ( ran  F  =  ran  ( A  X.  { B }
)  <->  ran  F  =  { B } ) )
41, 3syl5ib 219 . . 3  |-  ( A  =/=  (/)  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
54adantl 466 . 2  |-  ( ( F  Fn  A  /\  A  =/=  (/) )  ->  ( F  =  ( A  X.  { B } )  ->  ran  F  =  { B } ) )
6 df-fo 5592 . . . . . . 7  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
7 fof 5793 . . . . . . 7  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
86, 7sylbir 213 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
9 fconst2g 6113 . . . . . 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 436 . . . 4  |-  ( B  e.  _V  ->  ( F  Fn  A  ->  ( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
1211adantrd 468 . . 3  |-  ( B  e.  _V  ->  (
( F  Fn  A  /\  A  =/=  (/) )  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
13 fnrel 5677 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
14 snprc 4091 . . . . . 6  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
15 relrn0 5258 . . . . . . . . . 10  |-  ( Rel 
F  ->  ( F  =  (/)  <->  ran  F  =  (/) ) )
1615biimprd 223 . . . . . . . . 9  |-  ( Rel 
F  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
1716adantl 466 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  (/)  ->  F  =  (/) ) )
18 eqeq2 2482 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
1918adantr 465 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
20 xpeq2 5014 . . . . . . . . . . 11  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  ( A  X.  (/) ) )
21 xp0 5423 . . . . . . . . . . 11  |-  ( A  X.  (/) )  =  (/)
2220, 21syl6eq 2524 . . . . . . . . . 10  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  (/) )
2322eqeq2d 2481 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( F  =  ( A  X.  { B } )  <->  F  =  (/) ) )
2423adantr 465 . . . . . . . 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 434 . . . . . 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 32 . . . 4  |-  ( -.  B  e.  _V  ->  ( F  Fn  A  -> 
( ran  F  =  { B }  ->  F  =  ( A  X.  { B } ) ) ) )
2928adantrd 468 . . 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 191 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 369    = wceq 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3113   (/)c0 3785   {csn 4027    X. cxp 4997   ran crn 5000   Rel wrel 5004    Fn wfn 5581   -->wf 5582   -onto->wfo 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fo 5592  df-fv 5594
This theorem is referenced by:  nvo00  25352  esumnul  27699  esum0  27700  volsupnfl  29636  rnmptc  31027
  Copyright terms: Public domain W3C validator