ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfmpt2 Structured version   Unicode version

Theorem dfmpt2 5786
Description: Alternate definition for the "maps to" notation df-mpt2 5460 (although it requires that  C be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpt2.1  C 
_V
Assertion
Ref Expression
dfmpt2  ,  |->  C  U_  U_  { <. <. ,  >. ,  C >. }
Distinct variable groups:   ,,   ,,
Allowed substitution hints:    C(,)

Proof of Theorem dfmpt2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 mpt2mpts 5766 . 2  ,  |->  C  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C
2 vex 2554 . . . . 5 
_V
3 1stexg 5736 . . . . 5  _V  1st ` 
_V
42, 3ax-mp 7 . . . 4  1st `  _V
5 2ndexg 5737 . . . . . 6  _V  2nd ` 
_V
62, 5ax-mp 7 . . . . 5  2nd `  _V
7 dfmpt2.1 . . . . 5  C 
_V
86, 7csbexa 3877 . . . 4  [_ 2nd `  ]_ C  _V
94, 8csbexa 3877 . . 3  [_ 1st `  ]_ [_ 2nd `  ]_ C  _V
109dfmpt 5283 . 2  X.  |->  [_ 1st `  ]_ [_ 2nd `  ]_ C  U_  X.  { <. ,  [_ 1st `  ]_
[_ 2nd `  ]_ C >. }
11 nfcv 2175 . . . . 5  F/_
12 nfcsb1v 2876 . . . . 5  F/_ [_ 1st `  ]_ [_ 2nd `  ]_ C
1311, 12nfop 3556 . . . 4  F/_ <. ,  [_ 1st `  ]_ [_ 2nd `  ]_ C >.
1413nfsn 3421 . . 3  F/_ { <. ,  [_ 1st `  ]_ [_ 2nd `  ]_ C >. }
15 nfcv 2175 . . . . 5  F/_
16 nfcv 2175 . . . . . 6  F/_ 1st `
17 nfcsb1v 2876 . . . . . 6  F/_ [_ 2nd `  ]_ C
1816, 17nfcsb 2878 . . . . 5  F/_ [_ 1st `  ]_ [_ 2nd `  ]_ C
1915, 18nfop 3556 . . . 4  F/_ <. ,  [_ 1st `  ]_ [_ 2nd `  ]_ C >.
2019nfsn 3421 . . 3  F/_ { <. ,  [_ 1st `  ]_ [_ 2nd `  ]_ C >. }
21 nfcv 2175 . . 3  F/_ { <. <. , 
>. ,  C >. }
22 id 19 . . . . 5  <. , 
>.  <. ,  >.
23 csbopeq1a 5756 . . . . 5  <. , 
>.  [_ 1st `  ]_
[_ 2nd `  ]_ C  C
2422, 23opeq12d 3548 . . . 4  <. , 
>.  <. ,  [_ 1st `  ]_ [_ 2nd `  ]_ C >.  <. <. ,  >. ,  C >.
2524sneqd 3380 . . 3  <. , 
>.  { <. ,  [_ 1st `  ]_ [_ 2nd `  ]_ C >. }  { <. <. , 
>. ,  C >. }
2614, 20, 21, 25iunxpf 4427 . 2  U_  X.  { <. ,  [_ 1st `  ]_ [_ 2nd `  ]_ C >. }  U_  U_  { <. <. ,  >. ,  C >. }
271, 10, 263eqtri 2061 1  ,  |->  C  U_  U_  { <. <. ,  >. ,  C >. }
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390   _Vcvv 2551   [_csb 2846   {csn 3367   <.cop 3370   U_ciun 3648    |-> cmpt 3809    X. cxp 4286   ` cfv 4845    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708
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-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-bnd 1396  ax-4 1397  ax-13 1401  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  ax-un 4136
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-ral 2305  df-rex 2306  df-reu 2307  df-v 2553  df-sbc 2759  df-csb 2847  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-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator