ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  2ndvalg Unicode version

Theorem 2ndvalg 6122
Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
2ndvalg  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )

Proof of Theorem 2ndvalg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snexg 4170 . . 3  |-  ( A  e.  _V  ->  { A }  e.  _V )
2 rnexg 4876 . . 3  |-  ( { A }  e.  _V  ->  ran  { A }  e.  _V )
3 uniexg 4424 . . 3  |-  ( ran 
{ A }  e.  _V  ->  U. ran  { A }  e.  _V )
41, 2, 33syl 17 . 2  |-  ( A  e.  _V  ->  U. ran  { A }  e.  _V )
5 sneq 3594 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
65rneqd 4840 . . . 4  |-  ( x  =  A  ->  ran  { x }  =  ran  { A } )
76unieqd 3807 . . 3  |-  ( x  =  A  ->  U. ran  { x }  =  U. ran  { A } )
8 df-2nd 6120 . . 3  |-  2nd  =  ( x  e.  _V  |->  U.
ran  { x } )
97, 8fvmptg 5572 . 2  |-  ( ( A  e.  _V  /\  U.
ran  { A }  e.  _V )  ->  ( 2nd `  A )  =  U. ran  { A } )
104, 9mpdan 419 1  |-  ( A  e.  _V  ->  ( 2nd `  A )  = 
U. ran  { A } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348    e. wcel 2141   _Vcvv 2730   {csn 3583   U.cuni 3796   ran crn 4612   ` cfv 5198   2ndc2nd 6118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-2nd 6120
This theorem is referenced by:  2nd0  6124  op2nd  6126  elxp6  6148
  Copyright terms: Public domain W3C validator