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Theorem xpsfrnel 14818
Description: Elementhood in the target space of the function  F appearing in xpsval 14827. (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 7473 . 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 980 . . 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 7143 . . . . . . . 8  |-  2o  =  { (/) ,  1o }
43raleqi 3062 . . . . . . 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 4577 . . . . . . . 8  |-  (/)  e.  _V
6 1on 7137 . . . . . . . . 9  |-  1o  e.  On
76elexi 3123 . . . . . . . 8  |-  1o  e.  _V
8 fveq2 5866 . . . . . . . . 9  |-  ( k  =  (/)  ->  ( G `
 k )  =  ( G `  (/) ) )
9 iftrue 3945 . . . . . . . . 9  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  A )
108, 9eleq12d 2549 . . . . . . . 8  |-  ( k  =  (/)  ->  ( ( G `  k )  e.  if ( k  =  (/) ,  A ,  B )  <->  ( G `  (/) )  e.  A
) )
11 fveq2 5866 . . . . . . . . 9  |-  ( k  =  1o  ->  ( G `  k )  =  ( G `  1o ) )
12 1n0 7145 . . . . . . . . . . 11  |-  1o  =/=  (/)
13 neeq1 2748 . . . . . . . . . . 11  |-  ( k  =  1o  ->  (
k  =/=  (/)  <->  1o  =/=  (/) ) )
1412, 13mpbiri 233 . . . . . . . . . 10  |-  ( k  =  1o  ->  k  =/=  (/) )
15 ifnefalse 3951 . . . . . . . . . 10  |-  ( k  =/=  (/)  ->  if (
k  =  (/) ,  A ,  B )  =  B )
1614, 15syl 16 . . . . . . . . 9  |-  ( k  =  1o  ->  if ( k  =  (/) ,  A ,  B )  =  B )
1711, 16eleq12d 2549 . . . . . . . 8  |-  ( k  =  1o  ->  (
( G `  k
)  e.  if ( k  =  (/) ,  A ,  B )  <->  ( G `  1o )  e.  B
) )
185, 7, 10, 17ralpr 4080 . . . . . . 7  |-  ( A. k  e.  { (/) ,  1o }  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B )  <-> 
( ( G `  (/) )  e.  A  /\  ( G `  1o )  e.  B ) )
194, 18bitri 249 . . . . . 6  |-  ( A. k  e.  2o  ( G `  k )  e.  if ( k  =  (/) ,  A ,  B
)  <->  ( ( G `
 (/) )  e.  A  /\  ( G `  1o )  e.  B )
)
20 2onn 7289 . . . . . . . . . 10  |-  2o  e.  om
21 nnfi 7710 . . . . . . . . . 10  |-  ( 2o  e.  om  ->  2o  e.  Fin )
2220, 21ax-mp 5 . . . . . . . . 9  |-  2o  e.  Fin
23 fnfi 7798 . . . . . . . . 9  |-  ( ( G  Fn  2o  /\  2o  e.  Fin )  ->  G  e.  Fin )
2422, 23mpan2 671 . . . . . . . 8  |-  ( G  Fn  2o  ->  G  e.  Fin )
25 elex 3122 . . . . . . . 8  |-  ( G  e.  Fin  ->  G  e.  _V )
2624, 25syl 16 . . . . . . 7  |-  ( G  Fn  2o  ->  G  e.  _V )
2726biantrurd 508 . . . . . 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 260 . . . . 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 637 . . . 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 977 . . . 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 977 . . . 4  |-  ( ( G  Fn  2o  /\  ( G `  (/) )  e.  A  /\  ( G `
 1o )  e.  B )  <->  ( G  Fn  2o  /\  ( ( G `  (/) )  e.  A  /\  ( G `
 1o )  e.  B ) ) )
3229, 30, 313bitr4i 277 . . 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 249 . 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 249 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   (/)c0 3785   ifcif 3939   {cpr 4029   Oncon0 4878    Fn wfn 5583   ` cfv 5588   omcom 6684   1oc1o 7123   2oc2o 7124   X_cixp 7469   Fincfn 7516
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  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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-reu 2821  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-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  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 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-ixp 7470  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520
This theorem is referenced by:  xpsfrnel2  14820  xpsff1o  14823
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