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Theorem cantnflem4 7640
Description: Lemma for cantnf 7641. Complete the induction step of cantnflem3 7639. (Contributed by Mario Carneiro, 25-May-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( 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.1  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
cantnf.2  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
cantnf.3  |-  ( ph  -> 
(/)  e.  C )
cantnf.4  |-  X  = 
U. |^| { c  e.  On  |  C  e.  ( A  ^o  c
) }
cantnf.5  |-  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.6  |-  Y  =  ( 1st `  P
)
cantnf.7  |-  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.2 . . . 4  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
2 cantnfs.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  On )
3 cantnfs.1 . . . . . . . . . . . . 13  |-  S  =  dom  ( A CNF  B
)
4 cantnfs.3 . . . . . . . . . . . . 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.1 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
7 cantnf.3 . . . . . . . . . . . . 13  |-  ( ph  -> 
(/)  e.  C )
83, 2, 4, 5, 6, 1, 7cantnflem2 7638 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
9 eqid 2435 . . . . . . . . . . . . . 14  |-  X  =  X
10 eqid 2435 . . . . . . . . . . . . . 14  |-  Y  =  Y
11 eqid 2435 . . . . . . . . . . . . . 14  |-  Z  =  Z
129, 10, 113pm3.2i 1132 . . . . . . . . . . . . 13  |-  ( X  =  X  /\  Y  =  Y  /\  Z  =  Z )
13 cantnf.4 . . . . . . . . . . . . . 14  |-  X  = 
U. |^| { c  e.  On  |  C  e.  ( A  ^o  c
) }
14 cantnf.5 . . . . . . . . . . . . . 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.6 . . . . . . . . . . . . . 14  |-  Y  =  ( 1st `  P
)
16 cantnf.7 . . . . . . . . . . . . . 14  |-  Z  =  ( 2nd `  P
)
1713, 14, 15, 16oeeui 6837 . . . . . . . . . . . . 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 225 . . . . . . . . . . . 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 446 . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  On  /\  Y  e.  ( A 
\  1o )  /\  Z  e.  ( A  ^o  X ) ) )
2120simp1d 969 . . . . . . . . 9  |-  ( ph  ->  X  e.  On )
22 oecl 6773 . . . . . . . . 9  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
232, 21, 22syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
2420simp2d 970 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( A 
\  1o ) )
2524eldifad 3324 . . . . . . . . 9  |-  ( ph  ->  Y  e.  A )
26 onelon 4598 . . . . . . . . 9  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
272, 25, 26syl2anc 643 . . . . . . . 8  |-  ( ph  ->  Y  e.  On )
28 omcl 6772 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Y  e.  On )  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
2923, 27, 28syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
3020simp3d 971 . . . . . . . 8  |-  ( ph  ->  Z  e.  ( A  ^o  X ) )
31 onelon 4598 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Z  e.  ( A  ^o  X ) )  ->  Z  e.  On )
3223, 30, 31syl2anc 643 . . . . . . 7  |-  ( ph  ->  Z  e.  On )
33 oaword1 6787 . . . . . . 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 643 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  C_  ( (
( A  ^o  X
)  .o  Y )  +o  Z ) )
35 dif1o 6736 . . . . . . . . . . 11  |-  ( Y  e.  ( A  \  1o )  <->  ( Y  e.  A  /\  Y  =/=  (/) ) )
3635simprbi 451 . . . . . . . . . 10  |-  ( Y  e.  ( A  \  1o )  ->  Y  =/=  (/) )
3724, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
38 on0eln0 4628 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
3927, 38syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
4037, 39mpbird 224 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  Y )
41 omword1 6808 . . . . . . . 8  |-  ( ( ( ( A  ^o  X )  e.  On  /\  Y  e.  On )  /\  (/)  e.  Y )  ->  ( A  ^o  X )  C_  (
( A  ^o  X
)  .o  Y ) )
4223, 27, 40, 41syl21anc 1183 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  C_  ( ( A  ^o  X )  .o  Y ) )
4342, 30sseldd 3341 . . . . . 6  |-  ( ph  ->  Z  e.  ( ( A  ^o  X )  .o  Y ) )
4434, 43sseldd 3341 . . . . 5  |-  ( ph  ->  Z  e.  ( ( ( A  ^o  X
)  .o  Y )  +o  Z ) )
4519simprd 450 . . . . 5  |-  ( ph  ->  ( ( ( A  ^o  X )  .o  Y )  +o  Z
)  =  C )
4644, 45eleqtrd 2511 . . . 4  |-  ( ph  ->  Z  e.  C )
471, 46sseldd 3341 . . 3  |-  ( ph  ->  Z  e.  ran  ( A CNF  B ) )
483, 2, 4cantnff 7621 . . . 4  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
49 ffn 5583 . . . 4  |-  ( ( A CNF  B ) : S --> ( A  ^o  B )  ->  ( A CNF  B )  Fn  S
)
50 fvelrnb 5766 . . . 4  |-  ( ( A CNF  B )  Fn  S  ->  ( Z  e.  ran  ( A CNF  B
)  <->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z ) )
5148, 49, 503syl 19 . . 3  |-  ( ph  ->  ( Z  e.  ran  ( A CNF  B )  <->  E. g  e.  S  ( ( A CNF  B ) `
 g )  =  Z ) )
5247, 51mpbid 202 . 2  |-  ( ph  ->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z )
532adantr 452 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  A  e.  On )
544adantr 452 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  B  e.  On )
556adantr 452 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ( A  ^o  B ) )
561adantr 452 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  C_  ran  ( A CNF 
B ) )
577adantr 452 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  (/) 
e.  C )
58 simprl 733 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
g  e.  S )
59 simprr 734 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
( ( A CNF  B
) `  g )  =  Z )
60 eqid 2435 . . 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 7639 . 2  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ran  ( A CNF 
B ) )
6252, 61rexlimddv 2826 1  |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    \ cdif 3309    C_ wss 3312   (/)c0 3620   ifcif 3731   <.cop 3809   U.cuni 4007   |^|cint 4042   {copab 4257    e. cmpt 4258   Oncon0 4573   dom cdm 4870   ran crn 4871   iotacio 5408    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340   1oc1o 6709   2oc2o 6710    +o coa 6713    .o comu 6714    ^o coe 6715   CNF ccnf 7608
This theorem is referenced by:  cantnf  7641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-seqom 6697  df-1o 6716  df-2o 6717  df-oadd 6720  df-omul 6721  df-oexp 6722  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-cnf 7609
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