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Theorem ctiunctlemudc 11845
Description: Lemma for ctiunct 11848. (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 ) }
Assertion
Ref Expression
ctiunctlemudc  |-  ( ph  ->  A. n  e.  om DECID  n  e.  U )
Distinct variable groups:    x, A    n, F, x    z, F, x   
n, J, x    z, J    S, n    z, S    T, n    z, T    U, n    ph, x
Allowed substitution hints:    ph( z, n)    A( z, n)    B( x, z, n)    S( x)    T( x)    U( x, z)    G( x, z, n)

Proof of Theorem ctiunctlemudc
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 eleq1 2178 . . . . . . . . 9  |-  ( n  =  ( 1st `  ( J `  m )
)  ->  ( n  e.  S  <->  ( 1st `  ( J `  m )
)  e.  S ) )
21dcbid 806 . . . . . . . 8  |-  ( n  =  ( 1st `  ( J `  m )
)  ->  (DECID  n  e.  S 
<-> DECID  ( 1st `  ( J `
 m ) )  e.  S ) )
3 ctiunct.sdc . . . . . . . . 9  |-  ( ph  ->  A. n  e.  om DECID  n  e.  S )
43adantr 272 . . . . . . . 8  |-  ( (
ph  /\  m  e.  om )  ->  A. n  e.  om DECID  n  e.  S )
5 ctiunct.j . . . . . . . . . . . 12  |-  ( ph  ->  J : om -1-1-onto-> ( om  X.  om ) )
65adantr 272 . . . . . . . . . . 11  |-  ( (
ph  /\  m  e.  om )  ->  J : om
-1-1-onto-> ( om  X.  om )
)
7 f1of 5333 . . . . . . . . . . 11  |-  ( J : om -1-1-onto-> ( om  X.  om )  ->  J : om --> ( om  X.  om )
)
86, 7syl 14 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  om )  ->  J : om
--> ( om  X.  om ) )
9 simpr 109 . . . . . . . . . 10  |-  ( (
ph  /\  m  e.  om )  ->  m  e.  om )
108, 9ffvelrnd 5522 . . . . . . . . 9  |-  ( (
ph  /\  m  e.  om )  ->  ( J `  m )  e.  ( om  X.  om )
)
11 xp1st 6029 . . . . . . . . 9  |-  ( ( J `  m )  e.  ( om  X.  om )  ->  ( 1st `  ( J `  m
) )  e.  om )
1210, 11syl 14 . . . . . . . 8  |-  ( (
ph  /\  m  e.  om )  ->  ( 1st `  ( J `  m
) )  e.  om )
132, 4, 12rspcdva 2766 . . . . . . 7  |-  ( (
ph  /\  m  e.  om )  -> DECID  ( 1st `  ( J `  m )
)  e.  S )
1413adantr 272 . . . . . 6  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  -> DECID  ( 1st `  ( J `  m )
)  e.  S )
15 eleq1 2178 . . . . . . . 8  |-  ( n  =  ( 2nd `  ( J `  m )
)  ->  ( n  e.  [_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T 
<->  ( 2nd `  ( J `  m )
)  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T ) )
1615dcbid 806 . . . . . . 7  |-  ( n  =  ( 2nd `  ( J `  m )
)  ->  (DECID  n  e.  [_ ( F `  ( 1st `  ( J `  m ) ) )  /  x ]_ T  <-> DECID  ( 2nd `  ( J `  m
) )  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T ) )
17 ctiunct.f . . . . . . . . . . 11  |-  ( ph  ->  F : S -onto-> A
)
18 fof 5313 . . . . . . . . . . 11  |-  ( F : S -onto-> A  ->  F : S --> A )
1917, 18syl 14 . . . . . . . . . 10  |-  ( ph  ->  F : S --> A )
2019ad2antrr 477 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  ->  F : S --> A )
21 simpr 109 . . . . . . . . 9  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  ->  ( 1st `  ( J `  m ) )  e.  S )
2220, 21ffvelrnd 5522 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  ->  ( F `  ( 1st `  ( J `  m
) ) )  e.  A )
23 ctiunct.tdc . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  A )  ->  A. n  e.  om DECID  n  e.  T )
2423ralrimiva 2480 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  A. n  e.  om DECID  n  e.  T )
2524ad2antrr 477 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  ->  A. x  e.  A  A. n  e.  om DECID  n  e.  T )
26 nfcv 2256 . . . . . . . . . 10  |-  F/_ x om
27 nfcsb1v 3003 . . . . . . . . . . . 12  |-  F/_ x [_ ( F `  ( 1st `  ( J `  m ) ) )  /  x ]_ T
2827nfcri 2250 . . . . . . . . . . 11  |-  F/ x  n  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T
2928nfdc 1620 . . . . . . . . . 10  |-  F/ xDECID  n  e.  [_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T
3026, 29nfralya 2448 . . . . . . . . 9  |-  F/ x A. n  e.  om DECID  n  e.  [_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T
31 csbeq1a 2981 . . . . . . . . . . . 12  |-  ( x  =  ( F `  ( 1st `  ( J `
 m ) ) )  ->  T  =  [_ ( F `  ( 1st `  ( J `  m ) ) )  /  x ]_ T
)
3231eleq2d 2185 . . . . . . . . . . 11  |-  ( x  =  ( F `  ( 1st `  ( J `
 m ) ) )  ->  ( n  e.  T  <->  n  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T ) )
3332dcbid 806 . . . . . . . . . 10  |-  ( x  =  ( F `  ( 1st `  ( J `
 m ) ) )  ->  (DECID  n  e.  T 
<-> DECID  n  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) )
3433ralbidv 2412 . . . . . . . . 9  |-  ( x  =  ( F `  ( 1st `  ( J `
 m ) ) )  ->  ( A. n  e.  om DECID  n  e.  T  <->  A. n  e.  om DECID  n  e.  [_ ( F `  ( 1st `  ( J `  m ) ) )  /  x ]_ T
) )
3530, 34rspc 2755 . . . . . . . 8  |-  ( ( F `  ( 1st `  ( J `  m
) ) )  e.  A  ->  ( A. x  e.  A  A. n  e.  om DECID  n  e.  T  ->  A. n  e.  om DECID  n  e.  [_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T ) )
3622, 25, 35sylc 62 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  ->  A. n  e.  om DECID  n  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T )
3710adantr 272 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  ->  ( J `  m )  e.  ( om  X.  om ) )
38 xp2nd 6030 . . . . . . . 8  |-  ( ( J `  m )  e.  ( om  X.  om )  ->  ( 2nd `  ( J `  m
) )  e.  om )
3937, 38syl 14 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  ->  ( 2nd `  ( J `  m ) )  e. 
om )
4016, 36, 39rspcdva 2766 . . . . . 6  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  -> DECID  ( 2nd `  ( J `  m )
)  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T )
41 dcan 901 . . . . . 6  |-  (DECID  ( 1st `  ( J `  m
) )  e.  S  ->  (DECID  ( 2nd `  ( J `  m )
)  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T  -> DECID  ( ( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) ) )
4214, 40, 41sylc 62 . . . . 5  |-  ( ( ( ph  /\  m  e.  om )  /\  ( 1st `  ( J `  m ) )  e.  S )  -> DECID  ( ( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) )
43 simpr 109 . . . . . . . 8  |-  ( ( ( ph  /\  m  e.  om )  /\  -.  ( 1st `  ( J `
 m ) )  e.  S )  ->  -.  ( 1st `  ( J `  m )
)  e.  S )
4443intnanrd 900 . . . . . . 7  |-  ( ( ( ph  /\  m  e.  om )  /\  -.  ( 1st `  ( J `
 m ) )  e.  S )  ->  -.  ( ( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) )
4544olcd 706 . . . . . 6  |-  ( ( ( ph  /\  m  e.  om )  /\  -.  ( 1st `  ( J `
 m ) )  e.  S )  -> 
( ( ( 1st `  ( J `  m
) )  e.  S  /\  ( 2nd `  ( J `  m )
)  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T )  \/ 
-.  ( ( 1st `  ( J `  m
) )  e.  S  /\  ( 2nd `  ( J `  m )
)  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T ) ) )
46 df-dc 803 . . . . . 6  |-  (DECID  ( ( 1st `  ( J `
 m ) )  e.  S  /\  ( 2nd `  ( J `  m ) )  e. 
[_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T )  <->  ( (
( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T )  \/  -.  ( ( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) ) )
4745, 46sylibr 133 . . . . 5  |-  ( ( ( ph  /\  m  e.  om )  /\  -.  ( 1st `  ( J `
 m ) )  e.  S )  -> DECID  (
( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) )
48 exmiddc 804 . . . . . 6  |-  (DECID  ( 1st `  ( J `  m
) )  e.  S  ->  ( ( 1st `  ( J `  m )
)  e.  S  \/  -.  ( 1st `  ( J `  m )
)  e.  S ) )
4913, 48syl 14 . . . . 5  |-  ( (
ph  /\  m  e.  om )  ->  ( ( 1st `  ( J `  m ) )  e.  S  \/  -.  ( 1st `  ( J `  m ) )  e.  S ) )
5042, 47, 49mpjaodan 770 . . . 4  |-  ( (
ph  /\  m  e.  om )  -> DECID  ( ( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) )
51 ibar 297 . . . . . . 7  |-  ( m  e.  om  ->  (
( ( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T )  <->  ( m  e.  om  /\  ( ( 1st `  ( J `
 m ) )  e.  S  /\  ( 2nd `  ( J `  m ) )  e. 
[_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T ) ) ) )
5251adantl 273 . . . . . 6  |-  ( (
ph  /\  m  e.  om )  ->  ( (
( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T )  <->  ( m  e.  om  /\  ( ( 1st `  ( J `
 m ) )  e.  S  /\  ( 2nd `  ( J `  m ) )  e. 
[_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T ) ) ) )
53 2fveq3 5392 . . . . . . . . 9  |-  ( z  =  m  ->  ( 1st `  ( J `  z ) )  =  ( 1st `  ( J `  m )
) )
5453eleq1d 2184 . . . . . . . 8  |-  ( z  =  m  ->  (
( 1st `  ( J `  z )
)  e.  S  <->  ( 1st `  ( J `  m
) )  e.  S
) )
55 2fveq3 5392 . . . . . . . . 9  |-  ( z  =  m  ->  ( 2nd `  ( J `  z ) )  =  ( 2nd `  ( J `  m )
) )
5653fveq2d 5391 . . . . . . . . . 10  |-  ( z  =  m  ->  ( F `  ( 1st `  ( J `  z
) ) )  =  ( F `  ( 1st `  ( J `  m ) ) ) )
5756csbeq1d 2979 . . . . . . . . 9  |-  ( z  =  m  ->  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T  =  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T )
5855, 57eleq12d 2186 . . . . . . . 8  |-  ( z  =  m  ->  (
( 2nd `  ( J `  z )
)  e.  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T  <->  ( 2nd `  ( J `  m
) )  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T ) )
5954, 58anbi12d 462 . . . . . . 7  |-  ( z  =  m  ->  (
( ( 1st `  ( J `  z )
)  e.  S  /\  ( 2nd `  ( J `
 z ) )  e.  [_ ( F `
 ( 1st `  ( J `  z )
) )  /  x ]_ T )  <->  ( ( 1st `  ( J `  m ) )  e.  S  /\  ( 2nd `  ( J `  m
) )  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T ) ) )
60 ctiunct.u . . . . . . 7  |-  U  =  { z  e.  om  |  ( ( 1st `  ( J `  z
) )  e.  S  /\  ( 2nd `  ( J `  z )
)  e.  [_ ( F `  ( 1st `  ( J `  z
) ) )  /  x ]_ T ) }
6159, 60elrab2 2814 . . . . . 6  |-  ( m  e.  U  <->  ( m  e.  om  /\  ( ( 1st `  ( J `
 m ) )  e.  S  /\  ( 2nd `  ( J `  m ) )  e. 
[_ ( F `  ( 1st `  ( J `
 m ) ) )  /  x ]_ T ) ) )
6252, 61syl6rbbr 198 . . . . 5  |-  ( (
ph  /\  m  e.  om )  ->  ( m  e.  U  <->  ( ( 1st `  ( J `  m
) )  e.  S  /\  ( 2nd `  ( J `  m )
)  e.  [_ ( F `  ( 1st `  ( J `  m
) ) )  /  x ]_ T ) ) )
6362dcbid 806 . . . 4  |-  ( (
ph  /\  m  e.  om )  ->  (DECID  m  e.  U 
<-> DECID  ( ( 1st `  ( J `  m )
)  e.  S  /\  ( 2nd `  ( J `
 m ) )  e.  [_ ( F `
 ( 1st `  ( J `  m )
) )  /  x ]_ T ) ) )
6450, 63mpbird 166 . . 3  |-  ( (
ph  /\  m  e.  om )  -> DECID  m  e.  U
)
6564ralrimiva 2480 . 2  |-  ( ph  ->  A. m  e.  om DECID  m  e.  U )
66 eleq1 2178 . . . 4  |-  ( m  =  n  ->  (
m  e.  U  <->  n  e.  U ) )
6766dcbid 806 . . 3  |-  ( m  =  n  ->  (DECID  m  e.  U  <-> DECID  n  e.  U )
)
6867cbvralv 2629 . 2  |-  ( A. m  e.  om DECID  m  e.  U  <->  A. n  e.  om DECID  n  e.  U )
6965, 68sylib 121 1  |-  ( ph  ->  A. n  e.  om DECID  n  e.  U )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680  DECID wdc 802    = wceq 1314    e. wcel 1463   A.wral 2391   {crab 2395   [_csb 2973    C_ wss 3039   omcom 4472    X. cxp 4505   -->wf 5087   -onto->wfo 5089   -1-1-onto->wf1o 5090   ` cfv 5091   1stc1st 6002   2ndc2nd 6003
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-1st 6004  df-2nd 6005
This theorem is referenced by:  ctiunct  11848
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