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Theorem cantnflem4 8121
Description: Lemma for cantnf 8122. Complete the induction step of cantnflem3 8120. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.s  |-  S  =  dom  ( A CNF  B
)
cantnfs.a  |-  ( ph  ->  A  e.  On )
cantnfs.b  |-  ( ph  ->  B  e.  On )
oemapval.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
cantnf.c  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
cantnf.s  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
cantnf.e  |-  ( ph  -> 
(/)  e.  C )
cantnf.x  |-  X  = 
U. |^| { c  e.  On  |  C  e.  ( A  ^o  c
) }
cantnf.p  |-  P  =  ( iota d E. a  e.  On  E. b  e.  ( A  ^o  X ) ( d  =  <. a ,  b
>.  /\  ( ( ( A  ^o  X )  .o  a )  +o  b )  =  C ) )
cantnf.y  |-  Y  =  ( 1st `  P
)
cantnf.z  |-  Z  =  ( 2nd `  P
)
Assertion
Ref Expression
cantnflem4  |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
Distinct variable groups:    w, c, x, y, z, B    a,
b, c, d, w, x, y, z, C    A, a, b, c, d, w, x, y, z    T, c    S, c, x, y, z    x, Z, y, z    ph, x, y, z    w, Y, x, y, z    X, a, b, d, w, x, y, z
Allowed substitution hints:    ph( w, a, b, c, d)    B( a, b, d)    P( x, y, z, w, a, b, c, d)    S( w, a, b, d)    T( x, y, z, w, a, b, d)    X( c)    Y( a, b, c, d)    Z( w, a, b, c, d)

Proof of Theorem cantnflem4
Dummy variables  g 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnf.s . . . 4  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
2 cantnfs.a . . . . . . . . 9  |-  ( ph  ->  A  e.  On )
3 cantnfs.s . . . . . . . . . . . . 13  |-  S  =  dom  ( A CNF  B
)
4 cantnfs.b . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  On )
5 oemapval.t . . . . . . . . . . . . 13  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
6 cantnf.c . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
7 cantnf.e . . . . . . . . . . . . 13  |-  ( ph  -> 
(/)  e.  C )
83, 2, 4, 5, 6, 1, 7cantnflem2 8119 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
9 eqid 2467 . . . . . . . . . . . . . 14  |-  X  =  X
10 eqid 2467 . . . . . . . . . . . . . 14  |-  Y  =  Y
11 eqid 2467 . . . . . . . . . . . . . 14  |-  Z  =  Z
129, 10, 113pm3.2i 1174 . . . . . . . . . . . . 13  |-  ( X  =  X  /\  Y  =  Y  /\  Z  =  Z )
13 cantnf.x . . . . . . . . . . . . . 14  |-  X  = 
U. |^| { c  e.  On  |  C  e.  ( A  ^o  c
) }
14 cantnf.p . . . . . . . . . . . . . 14  |-  P  =  ( iota d E. a  e.  On  E. b  e.  ( A  ^o  X ) ( d  =  <. a ,  b
>.  /\  ( ( ( A  ^o  X )  .o  a )  +o  b )  =  C ) )
15 cantnf.y . . . . . . . . . . . . . 14  |-  Y  =  ( 1st `  P
)
16 cantnf.z . . . . . . . . . . . . . 14  |-  Z  =  ( 2nd `  P
)
1713, 14, 15, 16oeeui 7261 . . . . . . . . . . . . 13  |-  ( ( A  e.  ( On 
\  2o )  /\  C  e.  ( On  \  1o ) )  -> 
( ( ( X  e.  On  /\  Y  e.  ( A  \  1o )  /\  Z  e.  ( A  ^o  X ) )  /\  ( ( ( A  ^o  X
)  .o  Y )  +o  Z )  =  C )  <->  ( X  =  X  /\  Y  =  Y  /\  Z  =  Z ) ) )
1812, 17mpbiri 233 . . . . . . . . . . . 12  |-  ( ( A  e.  ( On 
\  2o )  /\  C  e.  ( On  \  1o ) )  -> 
( ( X  e.  On  /\  Y  e.  ( A  \  1o )  /\  Z  e.  ( A  ^o  X ) )  /\  ( ( ( A  ^o  X
)  .o  Y )  +o  Z )  =  C ) )
198, 18syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  e.  On  /\  Y  e.  ( A  \  1o )  /\  Z  e.  ( A  ^o  X ) )  /\  ( ( ( A  ^o  X
)  .o  Y )  +o  Z )  =  C ) )
2019simpld 459 . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  On  /\  Y  e.  ( A 
\  1o )  /\  Z  e.  ( A  ^o  X ) ) )
2120simp1d 1008 . . . . . . . . 9  |-  ( ph  ->  X  e.  On )
22 oecl 7197 . . . . . . . . 9  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
232, 21, 22syl2anc 661 . . . . . . . 8  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
2420simp2d 1009 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( A 
\  1o ) )
2524eldifad 3493 . . . . . . . . 9  |-  ( ph  ->  Y  e.  A )
26 onelon 4908 . . . . . . . . 9  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
272, 25, 26syl2anc 661 . . . . . . . 8  |-  ( ph  ->  Y  e.  On )
28 omcl 7196 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Y  e.  On )  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
2923, 27, 28syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
3020simp3d 1010 . . . . . . . 8  |-  ( ph  ->  Z  e.  ( A  ^o  X ) )
31 onelon 4908 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Z  e.  ( A  ^o  X ) )  ->  Z  e.  On )
3223, 30, 31syl2anc 661 . . . . . . 7  |-  ( ph  ->  Z  e.  On )
33 oaword1 7211 . . . . . . 7  |-  ( ( ( ( A  ^o  X )  .o  Y
)  e.  On  /\  Z  e.  On )  ->  ( ( A  ^o  X )  .o  Y
)  C_  ( (
( A  ^o  X
)  .o  Y )  +o  Z ) )
3429, 32, 33syl2anc 661 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  C_  ( (
( A  ^o  X
)  .o  Y )  +o  Z ) )
35 dif1o 7160 . . . . . . . . . . 11  |-  ( Y  e.  ( A  \  1o )  <->  ( Y  e.  A  /\  Y  =/=  (/) ) )
3635simprbi 464 . . . . . . . . . 10  |-  ( Y  e.  ( A  \  1o )  ->  Y  =/=  (/) )
3724, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
38 on0eln0 4938 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
3927, 38syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
4037, 39mpbird 232 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  Y )
41 omword1 7232 . . . . . . . 8  |-  ( ( ( ( A  ^o  X )  e.  On  /\  Y  e.  On )  /\  (/)  e.  Y )  ->  ( A  ^o  X )  C_  (
( A  ^o  X
)  .o  Y ) )
4223, 27, 40, 41syl21anc 1227 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  C_  ( ( A  ^o  X )  .o  Y ) )
4342, 30sseldd 3510 . . . . . 6  |-  ( ph  ->  Z  e.  ( ( A  ^o  X )  .o  Y ) )
4434, 43sseldd 3510 . . . . 5  |-  ( ph  ->  Z  e.  ( ( ( A  ^o  X
)  .o  Y )  +o  Z ) )
4519simprd 463 . . . . 5  |-  ( ph  ->  ( ( ( A  ^o  X )  .o  Y )  +o  Z
)  =  C )
4644, 45eleqtrd 2557 . . . 4  |-  ( ph  ->  Z  e.  C )
471, 46sseldd 3510 . . 3  |-  ( ph  ->  Z  e.  ran  ( A CNF  B ) )
483, 2, 4cantnff 8103 . . . 4  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
49 ffn 5736 . . . 4  |-  ( ( A CNF  B ) : S --> ( A  ^o  B )  ->  ( A CNF  B )  Fn  S
)
50 fvelrnb 5920 . . . 4  |-  ( ( A CNF  B )  Fn  S  ->  ( Z  e.  ran  ( A CNF  B
)  <->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z ) )
5148, 49, 503syl 20 . . 3  |-  ( ph  ->  ( Z  e.  ran  ( A CNF  B )  <->  E. g  e.  S  ( ( A CNF  B ) `
 g )  =  Z ) )
5247, 51mpbid 210 . 2  |-  ( ph  ->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z )
532adantr 465 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  A  e.  On )
544adantr 465 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  B  e.  On )
556adantr 465 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ( A  ^o  B ) )
561adantr 465 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  C_  ran  ( A CNF 
B ) )
577adantr 465 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  (/) 
e.  C )
58 simprl 755 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
g  e.  S )
59 simprr 756 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
( ( A CNF  B
) `  g )  =  Z )
60 eqid 2467 . . 3  |-  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( g `  t ) ) )  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( g `  t ) ) )
613, 53, 54, 5, 55, 56, 57, 13, 14, 15, 16, 58, 59, 60cantnflem3 8120 . 2  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ran  ( A CNF 
B ) )
6252, 61rexlimddv 2963 1  |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818   {crab 2821    \ cdif 3478    C_ wss 3481   (/)c0 3790   ifcif 3944   <.cop 4038   U.cuni 4250   |^|cint 4287   {copab 4509    |-> cmpt 4510   Oncon0 4883   dom cdm 5004   ran crn 5005   iotacio 5554    Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294   1stc1st 6792   2ndc2nd 6793   1oc1o 7133   2oc2o 7134    +o coa 7137    .o comu 7138    ^o coe 7139   CNF ccnf 8088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-om 6695  df-1st 6794  df-2nd 6795  df-supp 6912  df-recs 7052  df-rdg 7086  df-seqom 7123  df-1o 7140  df-2o 7141  df-oadd 7144  df-omul 7145  df-oexp 7146  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-fsupp 7840  df-oi 7945  df-cnf 8089
This theorem is referenced by:  cantnf  8122
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