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Theorem xpsfrnel 15420
Description: Elementhood in the target space of the function  F appearing in xpsval 15429. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
xpsfrnel  |-  ( G  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
( G  Fn  2o  /\  ( G `  (/) )  e.  A  /\  ( G `
 1o )  e.  B ) )
Distinct variable groups:    A, k    B, k    k, G

Proof of Theorem xpsfrnel
StepHypRef Expression
1 elixp2 7534 . 2  |-  ( G  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
( G  e.  _V  /\  G  Fn  2o  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
) ) )
2 3ancoma 989 . . 3  |-  ( ( G  e.  _V  /\  G  Fn  2o  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
) )  <->  ( G  Fn  2o  /\  G  e. 
_V  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B ) ) )
3 df2o3 7203 . . . . . . . 8  |-  2o  =  { (/) ,  1o }
43raleqi 3036 . . . . . . 7  |-  ( A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
)  <->  A. k  e.  { (/)
,  1o }  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
) )
5 0ex 4557 . . . . . . . 8  |-  (/)  e.  _V
6 1on 7197 . . . . . . . . 9  |-  1o  e.  On
76elexi 3097 . . . . . . . 8  |-  1o  e.  _V
8 fveq2 5881 . . . . . . . . 9  |-  ( k  =  (/)  ->  ( G `
 k )  =  ( G `  (/) ) )
9 iftrue 3921 . . . . . . . . 9  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  A )
108, 9eleq12d 2511 . . . . . . . 8  |-  ( k  =  (/)  ->  ( ( G `  k )  e.  if ( k  =  (/) ,  A ,  B )  <->  ( G `  (/) )  e.  A
) )
11 fveq2 5881 . . . . . . . . 9  |-  ( k  =  1o  ->  ( G `  k )  =  ( G `  1o ) )
12 1n0 7205 . . . . . . . . . . 11  |-  1o  =/=  (/)
13 neeq1 2712 . . . . . . . . . . 11  |-  ( k  =  1o  ->  (
k  =/=  (/)  <->  1o  =/=  (/) ) )
1412, 13mpbiri 236 . . . . . . . . . 10  |-  ( k  =  1o  ->  k  =/=  (/) )
15 ifnefalse 3927 . . . . . . . . . 10  |-  ( k  =/=  (/)  ->  if (
k  =  (/) ,  A ,  B )  =  B )
1614, 15syl 17 . . . . . . . . 9  |-  ( k  =  1o  ->  if ( k  =  (/) ,  A ,  B )  =  B )
1711, 16eleq12d 2511 . . . . . . . 8  |-  ( k  =  1o  ->  (
( G `  k
)  e.  if ( k  =  (/) ,  A ,  B )  <->  ( G `  1o )  e.  B
) )
185, 7, 10, 17ralpr 4056 . . . . . . 7  |-  ( A. k  e.  { (/) ,  1o }  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B )  <-> 
( ( G `  (/) )  e.  A  /\  ( G `  1o )  e.  B ) )
194, 18bitri 252 . . . . . 6  |-  ( A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
)  <->  ( ( G `
 (/) )  e.  A  /\  ( G `  1o )  e.  B )
)
20 2onn 7349 . . . . . . . . . 10  |-  2o  e.  om
21 nnfi 7771 . . . . . . . . . 10  |-  ( 2o  e.  om  ->  2o  e.  Fin )
2220, 21ax-mp 5 . . . . . . . . 9  |-  2o  e.  Fin
23 fnfi 7855 . . . . . . . . 9  |-  ( ( G  Fn  2o  /\  2o  e.  Fin )  ->  G  e.  Fin )
2422, 23mpan2 675 . . . . . . . 8  |-  ( G  Fn  2o  ->  G  e.  Fin )
25 elex 3096 . . . . . . . 8  |-  ( G  e.  Fin  ->  G  e.  _V )
2624, 25syl 17 . . . . . . 7  |-  ( G  Fn  2o  ->  G  e.  _V )
2726biantrurd 510 . . . . . 6  |-  ( G  Fn  2o  ->  ( A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B )  <->  ( G  e.  _V  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B ) ) ) )
2819, 27syl5rbbr 263 . . . . 5  |-  ( G  Fn  2o  ->  (
( G  e.  _V  /\ 
A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B ) )  <->  ( ( G `  (/) )  e.  A  /\  ( G `
 1o )  e.  B ) ) )
2928pm5.32i 641 . . . 4  |-  ( ( G  Fn  2o  /\  ( G  e.  _V  /\ 
A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B ) ) )  <-> 
( G  Fn  2o  /\  ( ( G `  (/) )  e.  A  /\  ( G `  1o )  e.  B ) ) )
30 3anass 986 . . . 4  |-  ( ( G  Fn  2o  /\  G  e.  _V  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
) )  <->  ( G  Fn  2o  /\  ( G  e.  _V  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
) ) ) )
31 3anass 986 . . . 4  |-  ( ( G  Fn  2o  /\  ( G `  (/) )  e.  A  /\  ( G `
 1o )  e.  B )  <->  ( G  Fn  2o  /\  ( ( G `  (/) )  e.  A  /\  ( G `
 1o )  e.  B ) ) )
3229, 30, 313bitr4i 280 . . 3  |-  ( ( G  Fn  2o  /\  G  e.  _V  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
) )  <->  ( G  Fn  2o  /\  ( G `
 (/) )  e.  A  /\  ( G `  1o )  e.  B )
)
332, 32bitri 252 . 2  |-  ( ( G  e.  _V  /\  G  Fn  2o  /\  A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
) )  <->  ( G  Fn  2o  /\  ( G `
 (/) )  e.  A  /\  ( G `  1o )  e.  B )
)
341, 33bitri 252 1  |-  ( G  e.  X_ k  e.  2o  if ( k  =  (/) ,  A ,  B )  <-> 
( G  Fn  2o  /\  ( G `  (/) )  e.  A  /\  ( G `
 1o )  e.  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   _Vcvv 3087   (/)c0 3767   ifcif 3915   {cpr 4004   Oncon0 5442    Fn wfn 5596   ` cfv 5601   omcom 6706   1oc1o 7183   2oc2o 7184   X_cixp 7530   Fincfn 7577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581
This theorem is referenced by:  xpsfrnel2  15422  xpsff1o  15425
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