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Theorem imfstnrelc 25081
Description: The image under  1st of a class with no pairs inside. (Contributed by FL, 31-Aug-2009.)
Assertion
Ref Expression
imfstnrelc  |-  ( ( ( A  i^i  ( _V  X.  _V ) )  =  (/)  /\  A  =/=  (/) )  ->  ( 1st " A )  =  { (/)
} )

Proof of Theorem imfstnrelc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fo1st 6139 . . . . 5  |-  1st : _V -onto-> _V
2 fofun 5452 . . . . 5  |-  ( 1st
: _V -onto-> _V  ->  Fun 
1st )
31, 2ax-mp 8 . . . 4  |-  Fun  1st
4 ssv 3198 . . . . 5  |-  A  C_  _V
5 fofn 5453 . . . . . . 7  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
61, 5ax-mp 8 . . . . . 6  |-  1st  Fn  _V
7 fndm 5343 . . . . . 6  |-  ( 1st 
Fn  _V  ->  dom  1st  =  _V )
86, 7ax-mp 8 . . . . 5  |-  dom  1st  =  _V
94, 8sseqtr4i 3211 . . . 4  |-  A  C_  dom  1st
103, 9pm3.2i 441 . . 3  |-  ( Fun 
1st  /\  A  C_  dom  1st )
11 dfimafn2 5572 . . 3  |-  ( ( Fun  1st  /\  A  C_  dom  1st )  ->  ( 1st " A )  = 
U_ x  e.  A  { ( 1st `  x
) } )
1210, 11mp1i 11 . 2  |-  ( ( ( A  i^i  ( _V  X.  _V ) )  =  (/)  /\  A  =/=  (/) )  ->  ( 1st " A )  =  U_ x  e.  A  {
( 1st `  x
) } )
13 1stval 6124 . . . . 5  |-  ( 1st `  x )  =  U. dom  { x }
14 disj 3495 . . . . . . . . . . . 12  |-  ( ( A  i^i  ( _V 
X.  _V ) )  =  (/) 
<-> 
A. x  e.  A  -.  x  e.  ( _V  X.  _V ) )
1514biimpi 186 . . . . . . . . . . 11  |-  ( ( A  i^i  ( _V 
X.  _V ) )  =  (/)  ->  A. x  e.  A  -.  x  e.  ( _V  X.  _V ) )
16 df-ral 2548 . . . . . . . . . . 11  |-  ( A. x  e.  A  -.  x  e.  ( _V  X.  _V )  <->  A. x
( x  e.  A  ->  -.  x  e.  ( _V  X.  _V )
) )
1715, 16sylib 188 . . . . . . . . . 10  |-  ( ( A  i^i  ( _V 
X.  _V ) )  =  (/)  ->  A. x ( x  e.  A  ->  -.  x  e.  ( _V  X.  _V ) ) )
181719.21bi 1794 . . . . . . . . 9  |-  ( ( A  i^i  ( _V 
X.  _V ) )  =  (/)  ->  ( x  e.  A  ->  -.  x  e.  ( _V  X.  _V ) ) )
1918adantr 451 . . . . . . . 8  |-  ( ( ( A  i^i  ( _V  X.  _V ) )  =  (/)  /\  A  =/=  (/) )  ->  ( x  e.  A  ->  -.  x  e.  ( _V  X.  _V ) ) )
2019imp 418 . . . . . . 7  |-  ( ( ( ( A  i^i  ( _V  X.  _V )
)  =  (/)  /\  A  =/=  (/) )  /\  x  e.  A )  ->  -.  x  e.  ( _V  X.  _V ) )
21 dmsnn0 5138 . . . . . . . . . 10  |-  ( x  e.  ( _V  X.  _V )  <->  dom  { x }  =/=  (/) )
2221bicomi 193 . . . . . . . . 9  |-  ( dom 
{ x }  =/=  (/)  <->  x  e.  ( _V  X.  _V ) )
2322necon1bbii 2498 . . . . . . . 8  |-  ( -.  x  e.  ( _V 
X.  _V )  <->  dom  { x }  =  (/) )
24 0ss 3483 . . . . . . . . 9  |-  (/)  C_  { (/) }
25 sseq1 3199 . . . . . . . . 9  |-  ( dom 
{ x }  =  (/) 
->  ( dom  { x }  C_  { (/) }  <->  (/)  C_  { (/) } ) )
2624, 25mpbiri 224 . . . . . . . 8  |-  ( dom 
{ x }  =  (/) 
->  dom  { x }  C_ 
{ (/) } )
2723, 26sylbi 187 . . . . . . 7  |-  ( -.  x  e.  ( _V 
X.  _V )  ->  dom  { x }  C_  { (/) } )
2820, 27syl 15 . . . . . 6  |-  ( ( ( ( A  i^i  ( _V  X.  _V )
)  =  (/)  /\  A  =/=  (/) )  /\  x  e.  A )  ->  dom  { x }  C_  { (/) } )
29 uni0b 3852 . . . . . 6  |-  ( U. dom  { x }  =  (/)  <->  dom 
{ x }  C_  {
(/) } )
3028, 29sylibr 203 . . . . 5  |-  ( ( ( ( A  i^i  ( _V  X.  _V )
)  =  (/)  /\  A  =/=  (/) )  /\  x  e.  A )  ->  U. dom  { x }  =  (/) )
3113, 30syl5eq 2327 . . . 4  |-  ( ( ( ( A  i^i  ( _V  X.  _V )
)  =  (/)  /\  A  =/=  (/) )  /\  x  e.  A )  ->  ( 1st `  x )  =  (/) )
3231sneqd 3653 . . 3  |-  ( ( ( ( A  i^i  ( _V  X.  _V )
)  =  (/)  /\  A  =/=  (/) )  /\  x  e.  A )  ->  { ( 1st `  x ) }  =  { (/) } )
3332iuneq2dv 3926 . 2  |-  ( ( ( A  i^i  ( _V  X.  _V ) )  =  (/)  /\  A  =/=  (/) )  ->  U_ x  e.  A  { ( 1st `  x ) }  =  U_ x  e.  A  { (/) } )
34 iunconst 3913 . . 3  |-  ( A  =/=  (/)  ->  U_ x  e.  A  { (/) }  =  { (/) } )
3534adantl 452 . 2  |-  ( ( ( A  i^i  ( _V  X.  _V ) )  =  (/)  /\  A  =/=  (/) )  ->  U_ x  e.  A  { (/) }  =  { (/) } )
3612, 33, 353eqtrd 2319 1  |-  ( ( ( A  i^i  ( _V  X.  _V ) )  =  (/)  /\  A  =/=  (/) )  ->  ( 1st " A )  =  { (/)
} )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   U_ciun 3905    X. cxp 4687   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255   1stc1st 6120
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  ax-un 4512
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-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-iun 3907  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  df-1st 6122
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