ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ctiunctlemu2nd Unicode version

Theorem ctiunctlemu2nd 12390
Description: Lemma for ctiunct 12395. (Contributed by Jim Kingdon, 28-Oct-2023.)
Hypotheses
Ref Expression
ctiunct.som  |-  ( ph  ->  S  C_  om )
ctiunct.sdc  |-  ( ph  ->  A. n  e.  om DECID  n  e.  S )
ctiunct.f  |-  ( ph  ->  F : S -onto-> A
)
ctiunct.tom  |-  ( (
ph  /\  x  e.  A )  ->  T  C_ 
om )
ctiunct.tdc  |-  ( (
ph  /\  x  e.  A )  ->  A. n  e.  om DECID  n  e.  T )
ctiunct.g  |-  ( (
ph  /\  x  e.  A )  ->  G : T -onto-> B )
ctiunct.j  |-  ( ph  ->  J : om -1-1-onto-> ( om  X.  om ) )
ctiunct.u  |-  U  =  { z  e.  om  |  ( ( 1st `  ( J `  z
) )  e.  S  /\  ( 2nd `  ( J `  z )
)  e.  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T ) }
ctiunctlem.n  |-  ( ph  ->  N  e.  U )
Assertion
Ref Expression
ctiunctlemu2nd  |-  ( ph  ->  ( 2nd `  ( J `  N )
)  e.  [_ ( F `  ( 1st `  ( J `  N
) ) )  /  x ]_ T )
Distinct variable groups:    z, F    z, J    z, N    z, S    z, T    x, z
Allowed substitution hints:    ph( x, z, n)    A( x, z, n)    B( x, z, n)    S( x, n)    T( x, n)    U( x, z, n)    F( x, n)    G( x, z, n)    J( x, n)    N( x, n)

Proof of Theorem ctiunctlemu2nd
StepHypRef Expression
1 ctiunctlem.n . . . 4  |-  ( ph  ->  N  e.  U )
2 2fveq3 5501 . . . . . . 7  |-  ( z  =  N  ->  ( 1st `  ( J `  z ) )  =  ( 1st `  ( J `  N )
) )
32eleq1d 2239 . . . . . 6  |-  ( z  =  N  ->  (
( 1st `  ( J `  z )
)  e.  S  <->  ( 1st `  ( J `  N
) )  e.  S
) )
4 2fveq3 5501 . . . . . . 7  |-  ( z  =  N  ->  ( 2nd `  ( J `  z ) )  =  ( 2nd `  ( J `  N )
) )
52fveq2d 5500 . . . . . . . 8  |-  ( z  =  N  ->  ( F `  ( 1st `  ( J `  z
) ) )  =  ( F `  ( 1st `  ( J `  N ) ) ) )
65csbeq1d 3056 . . . . . . 7  |-  ( z  =  N  ->  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T  =  [_ ( F `  ( 1st `  ( J `  N
) ) )  /  x ]_ T )
74, 6eleq12d 2241 . . . . . 6  |-  ( z  =  N  ->  (
( 2nd `  ( J `  z )
)  e.  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T  <->  ( 2nd `  ( J `  N
) )  e.  [_ ( F `  ( 1st `  ( J `  N
) ) )  /  x ]_ T ) )
83, 7anbi12d 470 . . . . 5  |-  ( z  =  N  ->  (
( ( 1st `  ( J `  z )
)  e.  S  /\  ( 2nd `  ( J `
 z ) )  e.  [_ ( F `
 ( 1st `  ( J `  z )
) )  /  x ]_ T )  <->  ( ( 1st `  ( J `  N ) )  e.  S  /\  ( 2nd `  ( J `  N
) )  e.  [_ ( F `  ( 1st `  ( J `  N
) ) )  /  x ]_ T ) ) )
9 ctiunct.u . . . . 5  |-  U  =  { z  e.  om  |  ( ( 1st `  ( J `  z
) )  e.  S  /\  ( 2nd `  ( J `  z )
)  e.  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T ) }
108, 9elrab2 2889 . . . 4  |-  ( N  e.  U  <->  ( N  e.  om  /\  ( ( 1st `  ( J `
 N ) )  e.  S  /\  ( 2nd `  ( J `  N ) )  e. 
[_ ( F `  ( 1st `  ( J `
 N ) ) )  /  x ]_ T ) ) )
111, 10sylib 121 . . 3  |-  ( ph  ->  ( N  e.  om  /\  ( ( 1st `  ( J `  N )
)  e.  S  /\  ( 2nd `  ( J `
 N ) )  e.  [_ ( F `
 ( 1st `  ( J `  N )
) )  /  x ]_ T ) ) )
1211simprd 113 . 2  |-  ( ph  ->  ( ( 1st `  ( J `  N )
)  e.  S  /\  ( 2nd `  ( J `
 N ) )  e.  [_ ( F `
 ( 1st `  ( J `  N )
) )  /  x ]_ T ) )
1312simprd 113 1  |-  ( ph  ->  ( 2nd `  ( J `  N )
)  e.  [_ ( F `  ( 1st `  ( J `  N
) ) )  /  x ]_ T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103  DECID wdc 829    = wceq 1348    e. wcel 2141   A.wral 2448   {crab 2452   [_csb 3049    C_ wss 3121   omcom 4574    X. cxp 4609   -onto->wfo 5196   -1-1-onto->wf1o 5197   ` cfv 5198   1stc1st 6117   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-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206
This theorem is referenced by:  ctiunctlemf  12393
  Copyright terms: Public domain W3C validator