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Theorem cantnflem4 8197
Description: Lemma for cantnf 8198. Complete the induction step of cantnflem3 8196. (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 8195 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
9 eqid 2451 . . . . . . . . . . . . . 14  |-  X  =  X
10 eqid 2451 . . . . . . . . . . . . . 14  |-  Y  =  Y
11 eqid 2451 . . . . . . . . . . . . . 14  |-  Z  =  Z
129, 10, 113pm3.2i 1186 . . . . . . . . . . . . 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 7303 . . . . . . . . . . . . 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 237 . . . . . . . . . . . 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 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  e.  On  /\  Y  e.  ( A  \  1o )  /\  Z  e.  ( A  ^o  X ) )  /\  ( ( ( A  ^o  X
)  .o  Y )  +o  Z )  =  C ) )
2019simpld 461 . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  On  /\  Y  e.  ( A 
\  1o )  /\  Z  e.  ( A  ^o  X ) ) )
2120simp1d 1020 . . . . . . . . 9  |-  ( ph  ->  X  e.  On )
22 oecl 7239 . . . . . . . . 9  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
232, 21, 22syl2anc 667 . . . . . . . 8  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
2420simp2d 1021 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( A 
\  1o ) )
2524eldifad 3416 . . . . . . . . 9  |-  ( ph  ->  Y  e.  A )
26 onelon 5448 . . . . . . . . 9  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
272, 25, 26syl2anc 667 . . . . . . . 8  |-  ( ph  ->  Y  e.  On )
28 omcl 7238 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Y  e.  On )  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
2923, 27, 28syl2anc 667 . . . . . . 7  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
3020simp3d 1022 . . . . . . . 8  |-  ( ph  ->  Z  e.  ( A  ^o  X ) )
31 onelon 5448 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Z  e.  ( A  ^o  X ) )  ->  Z  e.  On )
3223, 30, 31syl2anc 667 . . . . . . 7  |-  ( ph  ->  Z  e.  On )
33 oaword1 7253 . . . . . . 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 667 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  C_  ( (
( A  ^o  X
)  .o  Y )  +o  Z ) )
35 dif1o 7202 . . . . . . . . . . 11  |-  ( Y  e.  ( A  \  1o )  <->  ( Y  e.  A  /\  Y  =/=  (/) ) )
3635simprbi 466 . . . . . . . . . 10  |-  ( Y  e.  ( A  \  1o )  ->  Y  =/=  (/) )
3724, 36syl 17 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
38 on0eln0 5478 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
3927, 38syl 17 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
4037, 39mpbird 236 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  Y )
41 omword1 7274 . . . . . . . 8  |-  ( ( ( ( A  ^o  X )  e.  On  /\  Y  e.  On )  /\  (/)  e.  Y )  ->  ( A  ^o  X )  C_  (
( A  ^o  X
)  .o  Y ) )
4223, 27, 40, 41syl21anc 1267 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  C_  ( ( A  ^o  X )  .o  Y ) )
4342, 30sseldd 3433 . . . . . 6  |-  ( ph  ->  Z  e.  ( ( A  ^o  X )  .o  Y ) )
4434, 43sseldd 3433 . . . . 5  |-  ( ph  ->  Z  e.  ( ( ( A  ^o  X
)  .o  Y )  +o  Z ) )
4519simprd 465 . . . . 5  |-  ( ph  ->  ( ( ( A  ^o  X )  .o  Y )  +o  Z
)  =  C )
4644, 45eleqtrd 2531 . . . 4  |-  ( ph  ->  Z  e.  C )
471, 46sseldd 3433 . . 3  |-  ( ph  ->  Z  e.  ran  ( A CNF  B ) )
483, 2, 4cantnff 8179 . . . 4  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
49 ffn 5728 . . . 4  |-  ( ( A CNF  B ) : S --> ( A  ^o  B )  ->  ( A CNF  B )  Fn  S
)
50 fvelrnb 5912 . . . 4  |-  ( ( A CNF  B )  Fn  S  ->  ( Z  e.  ran  ( A CNF  B
)  <->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z ) )
5148, 49, 503syl 18 . . 3  |-  ( ph  ->  ( Z  e.  ran  ( A CNF  B )  <->  E. g  e.  S  ( ( A CNF  B ) `
 g )  =  Z ) )
5247, 51mpbid 214 . 2  |-  ( ph  ->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z )
532adantr 467 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  A  e.  On )
544adantr 467 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  B  e.  On )
556adantr 467 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ( A  ^o  B ) )
561adantr 467 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  C_  ran  ( A CNF 
B ) )
577adantr 467 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  (/) 
e.  C )
58 simprl 764 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
g  e.  S )
59 simprr 766 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
( ( A CNF  B
) `  g )  =  Z )
60 eqid 2451 . . 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 8196 . 2  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ran  ( A CNF 
B ) )
6252, 61rexlimddv 2883 1  |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741    \ cdif 3401    C_ wss 3404   (/)c0 3731   ifcif 3881   <.cop 3974   U.cuni 4198   |^|cint 4234   {copab 4460    |-> cmpt 4461   dom cdm 4834   ran crn 4835   Oncon0 5423   iotacio 5544    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290   1stc1st 6791   2ndc2nd 6792   1oc1o 7175   2oc2o 7176    +o coa 7179    .o comu 7180    ^o coe 7181   CNF ccnf 8166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-seqom 7165  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-oexp 7188  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-cnf 8167
This theorem is referenced by:  cantnf  8198
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