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Theorem fconst5 5950
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 5080 . . . 4  |-  ( F  =  ( A  X.  { B } )  ->  ran  F  =  ran  ( A  X.  { B }
) )
2 rnxp 5283 . . . . 5  |-  ( A  =/=  (/)  ->  ran  ( A  X.  { B }
)  =  { B } )
32eqeq2d 2454 . . . 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 5439 . . . . . . 7  |-  ( F : A -onto-> { B } 
<->  ( F  Fn  A  /\  ran  F  =  { B } ) )
7 fof 5635 . . . . . . 7  |-  ( F : A -onto-> { B }  ->  F : A --> { B } )
86, 7sylbir 213 . . . . . 6  |-  ( ( F  Fn  A  /\  ran  F  =  { B } )  ->  F : A --> { B }
)
9 fconst2g 5947 . . . . . 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 5524 . . . . 5  |-  ( F  Fn  A  ->  Rel  F )
14 snprc 3954 . . . . . 6  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
15 relrn0 5112 . . . . . . . . . 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 2452 . . . . . . . . 9  |-  ( { B }  =  (/)  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
1918adantr 465 . . . . . . . 8  |-  ( ( { B }  =  (/) 
/\  Rel  F )  ->  ( ran  F  =  { B }  <->  ran  F  =  (/) ) )
20 xpeq2 4870 . . . . . . . . . . 11  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  ( A  X.  (/) ) )
21 xp0 5271 . . . . . . . . . . 11  |-  ( A  X.  (/) )  =  (/)
2220, 21syl6eq 2491 . . . . . . . . . 10  |-  ( { B }  =  (/)  ->  ( A  X.  { B } )  =  (/) )
2322eqeq2d 2454 . . . . . . . . 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 1369    e. wcel 1756    =/= wne 2620   _Vcvv 2987   (/)c0 3652   {csn 3892    X. cxp 4853   ran crn 4856   Rel wrel 4860    Fn wfn 5428   -->wf 5429   -onto->wfo 5431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-fo 5439  df-fv 5441
This theorem is referenced by:  nvo00  24176  esumnul  26517  esum0  26518  volsupnfl  28455
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