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Theorem cantnflem4OLD 8124
Description: Lemma for cantnfOLD 8125. Complete the induction step of cantnflem3OLD 8123. (Contributed by Mario Carneiro, 25-May-2015.) Obsolete version of cantnflem4 8102 as of 2-Jul-2019. (New usage is discouraged.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
cantnfsOLD.1  |-  S  =  dom  ( A CNF  B
)
cantnfsOLD.2  |-  ( ph  ->  A  e.  On )
cantnfsOLD.3  |-  ( ph  ->  B  e.  On )
oemapvalOLD.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  B  ( (
x `  z )  e.  ( y `  z
)  /\  A. w  e.  B  ( z  e.  w  ->  ( x `
 w )  =  ( y `  w
) ) ) }
cantnfOLD.1  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
cantnfOLD.2  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
cantnfOLD.3  |-  ( ph  -> 
(/)  e.  C )
cantnfOLD.4  |-  X  = 
U. |^| { c  e.  On  |  C  e.  ( A  ^o  c
) }
cantnfOLD.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 ) )
cantnfOLD.6  |-  Y  =  ( 1st `  P
)
cantnfOLD.7  |-  Z  =  ( 2nd `  P
)
Assertion
Ref Expression
cantnflem4OLD  |-  ( 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 cantnflem4OLD
Dummy variables  g 
t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cantnfOLD.2 . . . 4  |-  ( ph  ->  C  C_  ran  ( A CNF 
B ) )
2 cantnfsOLD.2 . . . . . . . . 9  |-  ( ph  ->  A  e.  On )
3 cantnfsOLD.1 . . . . . . . . . . . . 13  |-  S  =  dom  ( A CNF  B
)
4 cantnfsOLD.3 . . . . . . . . . . . . 13  |-  ( ph  ->  B  e.  On )
5 oemapvalOLD.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 cantnfOLD.1 . . . . . . . . . . . . 13  |-  ( ph  ->  C  e.  ( A  ^o  B ) )
7 cantnfOLD.3 . . . . . . . . . . . . 13  |-  ( ph  -> 
(/)  e.  C )
83, 2, 4, 5, 6, 1, 7cantnflem2 8100 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
9 eqid 2454 . . . . . . . . . . . . . 14  |-  X  =  X
10 eqid 2454 . . . . . . . . . . . . . 14  |-  Y  =  Y
11 eqid 2454 . . . . . . . . . . . . . 14  |-  Z  =  Z
129, 10, 113pm3.2i 1172 . . . . . . . . . . . . 13  |-  ( X  =  X  /\  Y  =  Y  /\  Z  =  Z )
13 cantnfOLD.4 . . . . . . . . . . . . . 14  |-  X  = 
U. |^| { c  e.  On  |  C  e.  ( A  ^o  c
) }
14 cantnfOLD.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 cantnfOLD.6 . . . . . . . . . . . . . 14  |-  Y  =  ( 1st `  P
)
16 cantnfOLD.7 . . . . . . . . . . . . . 14  |-  Z  =  ( 2nd `  P
)
1713, 14, 15, 16oeeui 7243 . . . . . . . . . . . . 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 457 . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  On  /\  Y  e.  ( A 
\  1o )  /\  Z  e.  ( A  ^o  X ) ) )
2120simp1d 1006 . . . . . . . . 9  |-  ( ph  ->  X  e.  On )
22 oecl 7179 . . . . . . . . 9  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
232, 21, 22syl2anc 659 . . . . . . . 8  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
2420simp2d 1007 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( A 
\  1o ) )
2524eldifad 3473 . . . . . . . . 9  |-  ( ph  ->  Y  e.  A )
26 onelon 4892 . . . . . . . . 9  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
272, 25, 26syl2anc 659 . . . . . . . 8  |-  ( ph  ->  Y  e.  On )
28 omcl 7178 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Y  e.  On )  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
2923, 27, 28syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
3020simp3d 1008 . . . . . . . 8  |-  ( ph  ->  Z  e.  ( A  ^o  X ) )
31 onelon 4892 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Z  e.  ( A  ^o  X ) )  ->  Z  e.  On )
3223, 30, 31syl2anc 659 . . . . . . 7  |-  ( ph  ->  Z  e.  On )
33 oaword1 7193 . . . . . . 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 659 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  C_  ( (
( A  ^o  X
)  .o  Y )  +o  Z ) )
35 dif1o 7142 . . . . . . . . . . 11  |-  ( Y  e.  ( A  \  1o )  <->  ( Y  e.  A  /\  Y  =/=  (/) ) )
3635simprbi 462 . . . . . . . . . 10  |-  ( Y  e.  ( A  \  1o )  ->  Y  =/=  (/) )
3724, 36syl 16 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
38 on0eln0 4922 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
3927, 38syl 16 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
4037, 39mpbird 232 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  Y )
41 omword1 7214 . . . . . . . 8  |-  ( ( ( ( A  ^o  X )  e.  On  /\  Y  e.  On )  /\  (/)  e.  Y )  ->  ( A  ^o  X )  C_  (
( A  ^o  X
)  .o  Y ) )
4223, 27, 40, 41syl21anc 1225 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  C_  ( ( A  ^o  X )  .o  Y ) )
4342, 30sseldd 3490 . . . . . 6  |-  ( ph  ->  Z  e.  ( ( A  ^o  X )  .o  Y ) )
4434, 43sseldd 3490 . . . . 5  |-  ( ph  ->  Z  e.  ( ( ( A  ^o  X
)  .o  Y )  +o  Z ) )
4519simprd 461 . . . . 5  |-  ( ph  ->  ( ( ( A  ^o  X )  .o  Y )  +o  Z
)  =  C )
4644, 45eleqtrd 2544 . . . 4  |-  ( ph  ->  Z  e.  C )
471, 46sseldd 3490 . . 3  |-  ( ph  ->  Z  e.  ran  ( A CNF  B ) )
483, 2, 4cantnff 8084 . . . 4  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
49 ffn 5713 . . . 4  |-  ( ( A CNF  B ) : S --> ( A  ^o  B )  ->  ( A CNF  B )  Fn  S
)
50 fvelrnb 5895 . . . 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 463 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  A  e.  On )
544adantr 463 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  B  e.  On )
556adantr 463 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ( A  ^o  B ) )
561adantr 463 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  C_  ran  ( A CNF 
B ) )
577adantr 463 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  (/) 
e.  C )
58 simprl 754 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
g  e.  S )
59 simprr 755 . . 3  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
( ( A CNF  B
) `  g )  =  Z )
60 eqid 2454 . . 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, 60cantnflem3OLD 8123 . 2  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ran  ( A CNF 
B ) )
6252, 61rexlimddv 2950 1  |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808    \ cdif 3458    C_ wss 3461   (/)c0 3783   ifcif 3929   <.cop 4022   U.cuni 4235   |^|cint 4271   {copab 4496    |-> cmpt 4497   Oncon0 4867   dom cdm 4988   ran crn 4989   iotacio 5532    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   1oc1o 7115   2oc2o 7116    +o coa 7119    .o comu 7120    ^o coe 7121   CNF ccnf 8069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-seqom 7105  df-1o 7122  df-2o 7123  df-oadd 7126  df-omul 7127  df-oexp 7128  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-oi 7927  df-cnf 8070
This theorem is referenced by:  cantnfOLD  8125
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