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Theorem fnniniseg2 5233
Description: Support sets of functions expressed as abstractions. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnniniseg2  F  Fn  `' F " _V 
\  { }  {  |  F `  =/=  }
Distinct variable groups:   ,   , F   ,

Proof of Theorem fnniniseg2
StepHypRef Expression
1 fncnvima2 5231 . 2  F  Fn  `' F " _V 
\  { }  {  |  F `  _V  \  { } }
2 funfvex 5135 . . . . . 6  Fun  F  dom  F  F `  _V
32funfni 4942 . . . . 5  F  Fn  F `  _V
43biantrurd 289 . . . 4  F  Fn  F `  =/=  F `  _V  F `  =/=
5 eldifsn 3486 . . . 4  F `  _V  \  { }  F `  _V  F `  =/=
64, 5syl6rbbr 188 . . 3  F  Fn  F `  _V  \  { }  F `  =/=
76rabbidva 2542 . 2  F  Fn  {  |  F `  _V  \  { } }  {  |  F `  =/=  }
81, 7eqtrd 2069 1  F  Fn  `' F " _V 
\  { }  {  |  F `  =/=  }
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390    =/= wne 2201   {crab 2304   _Vcvv 2551    \ cdif 2908   {csn 3367   `'ccnv 4287   "cima 4291    Fn wfn 4840   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by: (None)
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