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Theorem f2ndres 5729
Description: Mapping of a restriction of the  2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
f2ndres  2nd  |`  X.  :  X.  -->

Proof of Theorem f2ndres
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . . 8 
_V
2 vex 2554 . . . . . . . 8 
_V
31, 2op2nda 4748 . . . . . . 7  U. ran  {
<. ,  >. }
43eleq1i 2100 . . . . . 6  U. ran  { <. , 
>. }
54biimpri 124 . . . . 5  U. ran  { <. , 
>. }
65adantl 262 . . . 4  U. ran  { <. ,  >. }
76rgen2 2399 . . 3  U. ran  { <. ,  >. }
8 sneq 3378 . . . . . . 7  <. , 
>.  { }  { <. , 
>. }
98rneqd 4506 . . . . . 6  <. , 
>.  ran  { }  ran  { <. ,  >. }
109unieqd 3582 . . . . 5  <. , 
>.  U. ran  { }  U. ran  { <. , 
>. }
1110eleq1d 2103 . . . 4  <. , 
>.  U. ran  { }  U. ran  { <. ,  >. }
1211ralxp 4422 . . 3  X.  U. ran  { }  U. ran  { <. , 
>. }
137, 12mpbir 134 . 2  X.  U. ran  { }
14 df-2nd 5710 . . . . 5  2nd  _V  |->  U.
ran  { }
1514reseq1i 4551 . . . 4  2nd  |`  X.  _V  |->  U.
ran  { }  |`  X.
16 ssv 2959 . . . . 5  X.  C_  _V
17 resmpt 4599 . . . . 5  X. 
C_  _V  _V  |->  U.
ran  { }  |`  X.  X.  |->  U. ran  { }
1816, 17ax-mp 7 . . . 4  _V  |->  U.
ran  { }  |`  X.  X.  |->  U. ran  { }
1915, 18eqtri 2057 . . 3  2nd  |`  X.  X.  |->  U. ran  { }
2019fmpt 5262 . 2  X.  U. ran  { }  2nd  |`  X.  :  X. 
-->
2113, 20mpbi 133 1  2nd  |`  X.  :  X.  -->
Colors of variables: wff set class
Syntax hints:   wceq 1242   wcel 1390  wral 2300   _Vcvv 2551    C_ wss 2911   {csn 3367   <.cop 3370   U.cuni 3571    |-> cmpt 3809    X. cxp 4286   ran crn 4289    |` cres 4290   -->wf 4841   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-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-ral 2305  df-rex 2306  df-rab 2309  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-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-2nd 5710
This theorem is referenced by:  fo2ndresm  5731  2ndcof  5733  f2ndf  5789
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