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

Theorem ctiunctlemf 11962
Description: Lemma for ctiunct 11964. (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 ) }
ctiunct.h  |-  H  =  ( n  e.  U  |->  ( [_ ( F `
 ( 1st `  ( J `  n )
) )  /  x ]_ G `  ( 2nd `  ( J `  n
) ) ) )
Assertion
Ref Expression
ctiunctlemf  |-  ( ph  ->  H : U --> U_ x  e.  A  B )
Distinct variable groups:    A, n, x    B, n    x, F, z   
x, J, z    z, S    z, T    U, n    ph, n, x    x, z, n
Allowed substitution hints:    ph( z)    A( z)    B( x, z)    S( x, n)    T( x, n)    U( x, z)    F( n)    G( x, z, n)    H( x, z, n)    J( n)

Proof of Theorem ctiunctlemf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 ctiunct.f . . . . . . . 8  |-  ( ph  ->  F : S -onto-> A
)
21adantr 274 . . . . . . 7  |-  ( (
ph  /\  n  e.  U )  ->  F : S -onto-> A )
3 fof 5345 . . . . . . 7  |-  ( F : S -onto-> A  ->  F : S --> A )
42, 3syl 14 . . . . . 6  |-  ( (
ph  /\  n  e.  U )  ->  F : S --> A )
5 ctiunct.som . . . . . . . 8  |-  ( ph  ->  S  C_  om )
65adantr 274 . . . . . . 7  |-  ( (
ph  /\  n  e.  U )  ->  S  C_ 
om )
7 ctiunct.sdc . . . . . . . 8  |-  ( ph  ->  A. n  e.  om DECID  n  e.  S )
87adantr 274 . . . . . . 7  |-  ( (
ph  /\  n  e.  U )  ->  A. n  e.  om DECID  n  e.  S )
9 ctiunct.tom . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  T  C_ 
om )
109adantlr 468 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  U )  /\  x  e.  A )  ->  T  C_ 
om )
11 ctiunct.tdc . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  A. n  e.  om DECID  n  e.  T )
1211adantlr 468 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  U )  /\  x  e.  A )  ->  A. n  e.  om DECID  n  e.  T )
13 ctiunct.g . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  G : T -onto-> B )
1413adantlr 468 . . . . . . 7  |-  ( ( ( ph  /\  n  e.  U )  /\  x  e.  A )  ->  G : T -onto-> B )
15 ctiunct.j . . . . . . . 8  |-  ( ph  ->  J : om -1-1-onto-> ( om  X.  om ) )
1615adantr 274 . . . . . . 7  |-  ( (
ph  /\  n  e.  U )  ->  J : om -1-1-onto-> ( om  X.  om ) )
17 ctiunct.u . . . . . . 7  |-  U  =  { z  e.  om  |  ( ( 1st `  ( J `  z
) )  e.  S  /\  ( 2nd `  ( J `  z )
)  e.  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T ) }
18 simpr 109 . . . . . . 7  |-  ( (
ph  /\  n  e.  U )  ->  n  e.  U )
196, 8, 2, 10, 12, 14, 16, 17, 18ctiunctlemu1st 11958 . . . . . 6  |-  ( (
ph  /\  n  e.  U )  ->  ( 1st `  ( J `  n ) )  e.  S )
204, 19ffvelrnd 5556 . . . . 5  |-  ( (
ph  /\  n  e.  U )  ->  ( F `  ( 1st `  ( J `  n
) ) )  e.  A )
21 fof 5345 . . . . . . . . . . 11  |-  ( G : T -onto-> B  ->  G : T --> B )
2213, 21syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  G : T --> B )
2322ralrimiva 2505 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  G : T --> B )
2423adantr 274 . . . . . . . 8  |-  ( (
ph  /\  n  e.  U )  ->  A. x  e.  A  G : T
--> B )
25 rspsbc 2991 . . . . . . . 8  |-  ( ( F `  ( 1st `  ( J `  n
) ) )  e.  A  ->  ( A. x  e.  A  G : T --> B  ->  [. ( F `  ( 1st `  ( J `  n
) ) )  /  x ]. G : T --> B ) )
2620, 24, 25sylc 62 . . . . . . 7  |-  ( (
ph  /\  n  e.  U )  ->  [. ( F `  ( 1st `  ( J `  n
) ) )  /  x ]. G : T --> B )
27 sbcfg 5271 . . . . . . . 8  |-  ( ( F `  ( 1st `  ( J `  n
) ) )  e.  A  ->  ( [. ( F `  ( 1st `  ( J `  n
) ) )  /  x ]. G : T --> B 
<-> 
[_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ G : [_ ( F `
 ( 1st `  ( J `  n )
) )  /  x ]_ T --> [_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ B ) )
2820, 27syl 14 . . . . . . 7  |-  ( (
ph  /\  n  e.  U )  ->  ( [. ( F `  ( 1st `  ( J `  n ) ) )  /  x ]. G : T --> B  <->  [_ ( F `
 ( 1st `  ( J `  n )
) )  /  x ]_ G : [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ T --> [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ B ) )
2926, 28mpbid 146 . . . . . 6  |-  ( (
ph  /\  n  e.  U )  ->  [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ G : [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ T --> [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ B )
306, 8, 2, 10, 12, 14, 16, 17, 18ctiunctlemu2nd 11959 . . . . . 6  |-  ( (
ph  /\  n  e.  U )  ->  ( 2nd `  ( J `  n ) )  e. 
[_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ T )
3129, 30ffvelrnd 5556 . . . . 5  |-  ( (
ph  /\  n  e.  U )  ->  ( [_ ( F `  ( 1st `  ( J `  n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n )
) )  e.  [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ B )
32 csbeq1 3006 . . . . . . 7  |-  ( y  =  ( F `  ( 1st `  ( J `
 n ) ) )  ->  [_ y  /  x ]_ B  =  [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ B )
3332eleq2d 2209 . . . . . 6  |-  ( y  =  ( F `  ( 1st `  ( J `
 n ) ) )  ->  ( ( [_ ( F `  ( 1st `  ( J `  n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n )
) )  e.  [_ y  /  x ]_ B  <->  (
[_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n
) ) )  e. 
[_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ B ) )
3433rspcev 2789 . . . . 5  |-  ( ( ( F `  ( 1st `  ( J `  n ) ) )  e.  A  /\  ( [_ ( F `  ( 1st `  ( J `  n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n )
) )  e.  [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ B )  ->  E. y  e.  A  ( [_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n
) ) )  e. 
[_ y  /  x ]_ B )
3520, 31, 34syl2anc 408 . . . 4  |-  ( (
ph  /\  n  e.  U )  ->  E. y  e.  A  ( [_ ( F `  ( 1st `  ( J `  n
) ) )  /  x ]_ G `  ( 2nd `  ( J `  n ) ) )  e.  [_ y  /  x ]_ B )
36 eliun 3817 . . . 4  |-  ( (
[_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n
) ) )  e. 
U_ y  e.  A  [_ y  /  x ]_ B 
<->  E. y  e.  A  ( [_ ( F `  ( 1st `  ( J `
 n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n
) ) )  e. 
[_ y  /  x ]_ B )
3735, 36sylibr 133 . . 3  |-  ( (
ph  /\  n  e.  U )  ->  ( [_ ( F `  ( 1st `  ( J `  n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n )
) )  e.  U_ y  e.  A  [_ y  /  x ]_ B )
38 nfcv 2281 . . . 4  |-  F/_ y B
39 nfcsb1v 3035 . . . 4  |-  F/_ x [_ y  /  x ]_ B
40 csbeq1a 3012 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
4138, 39, 40cbviun 3850 . . 3  |-  U_ x  e.  A  B  =  U_ y  e.  A  [_ y  /  x ]_ B
4237, 41eleqtrrdi 2233 . 2  |-  ( (
ph  /\  n  e.  U )  ->  ( [_ ( F `  ( 1st `  ( J `  n ) ) )  /  x ]_ G `  ( 2nd `  ( J `  n )
) )  e.  U_ x  e.  A  B
)
43 ctiunct.h . 2  |-  H  =  ( n  e.  U  |->  ( [_ ( F `
 ( 1st `  ( J `  n )
) )  /  x ]_ G `  ( 2nd `  ( J `  n
) ) ) )
4442, 43fmptd 5574 1  |-  ( ph  ->  H : U --> U_ x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104  DECID wdc 819    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   {crab 2420   [.wsbc 2909   [_csb 3003    C_ wss 3071   U_ciun 3813    |-> cmpt 3989   omcom 4504    X. cxp 4537   -->wf 5119   -onto->wfo 5121   -1-1-onto->wf1o 5122   ` cfv 5123   1stc1st 6036   2ndc2nd 6037
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131
This theorem is referenced by:  ctiunctlemfo  11963
  Copyright terms: Public domain W3C validator