MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsfrnel Structured version   Unicode version

Theorem xpsfrnel 14506
Description: Elementhood in the target space of the function  F appearing in xpsval 14515. (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 7272 . 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 972 . . 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 6938 . . . . . . . 8  |-  2o  =  { (/) ,  1o }
43raleqi 2926 . . . . . . 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 4427 . . . . . . . 8  |-  (/)  e.  _V
6 1on 6932 . . . . . . . . 9  |-  1o  e.  On
76elexi 2987 . . . . . . . 8  |-  1o  e.  _V
8 fveq2 5696 . . . . . . . . 9  |-  ( k  =  (/)  ->  ( G `
 k )  =  ( G `  (/) ) )
9 iftrue 3802 . . . . . . . . 9  |-  ( k  =  (/)  ->  if ( k  =  (/) ,  A ,  B )  =  A )
108, 9eleq12d 2511 . . . . . . . 8  |-  ( k  =  (/)  ->  ( ( G `  k )  e.  if ( k  =  (/) ,  A ,  B )  <->  ( G `  (/) )  e.  A
) )
11 fveq2 5696 . . . . . . . . 9  |-  ( k  =  1o  ->  ( G `  k )  =  ( G `  1o ) )
12 1n0 6940 . . . . . . . . . . 11  |-  1o  =/=  (/)
13 neeq1 2621 . . . . . . . . . . 11  |-  ( k  =  1o  ->  (
k  =/=  (/)  <->  1o  =/=  (/) ) )
1412, 13mpbiri 233 . . . . . . . . . 10  |-  ( k  =  1o  ->  k  =/=  (/) )
15 ifnefalse 3806 . . . . . . . . . 10  |-  ( k  =/=  (/)  ->  if (
k  =  (/) ,  A ,  B )  =  B )
1614, 15syl 16 . . . . . . . . 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 3934 . . . . . . 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 7084 . . . . . . . . . 10  |-  2o  e.  om
21 nnfi 7508 . . . . . . . . . 10  |-  ( 2o  e.  om  ->  2o  e.  Fin )
2220, 21ax-mp 5 . . . . . . . . 9  |-  2o  e.  Fin
23 fnfi 7594 . . . . . . . . 9  |-  ( ( G  Fn  2o  /\  2o  e.  Fin )  ->  G  e.  Fin )
2422, 23mpan2 671 . . . . . . . 8  |-  ( G  Fn  2o  ->  G  e.  Fin )
25 elex 2986 . . . . . . . 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 969 . . . 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 969 . . . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   _Vcvv 2977   (/)c0 3642   ifcif 3796   {cpr 3884   Oncon0 4724    Fn wfn 5418   ` cfv 5423   omcom 6481   1oc1o 6918   2oc2o 6919   X_cixp 7268   Fincfn 7315
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319
This theorem is referenced by:  xpsfrnel2  14508  xpsff1o  14511
  Copyright terms: Public domain W3C validator