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Theorem cantnflem4 7410
Description: Lemma for cantnf 7411. Complete the induction step of cantnflem3 7409. (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 7408 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  e.  ( On  \  2o )  /\  C  e.  ( On  \  1o ) ) )
9 eqid 2296 . . . . . . . . . . . . . 14  |-  X  =  X
10 eqid 2296 . . . . . . . . . . . . . 14  |-  Y  =  Y
11 eqid 2296 . . . . . . . . . . . . . 14  |-  Z  =  Z
129, 10, 113pm3.2i 1130 . . . . . . . . . . . . 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 6616 . . . . . . . . . . . . 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 224 . . . . . . . . . . . 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 15 . . . . . . . . . . 11  |-  ( ph  ->  ( ( X  e.  On  /\  Y  e.  ( A  \  1o )  /\  Z  e.  ( A  ^o  X ) )  /\  ( ( ( A  ^o  X
)  .o  Y )  +o  Z )  =  C ) )
2019simpld 445 . . . . . . . . . 10  |-  ( ph  ->  ( X  e.  On  /\  Y  e.  ( A 
\  1o )  /\  Z  e.  ( A  ^o  X ) ) )
2120simp1d 967 . . . . . . . . 9  |-  ( ph  ->  X  e.  On )
22 oecl 6552 . . . . . . . . 9  |-  ( ( A  e.  On  /\  X  e.  On )  ->  ( A  ^o  X
)  e.  On )
232, 21, 22syl2anc 642 . . . . . . . 8  |-  ( ph  ->  ( A  ^o  X
)  e.  On )
2420simp2d 968 . . . . . . . . . 10  |-  ( ph  ->  Y  e.  ( A 
\  1o ) )
25 eldifi 3311 . . . . . . . . . 10  |-  ( Y  e.  ( A  \  1o )  ->  Y  e.  A )
2624, 25syl 15 . . . . . . . . 9  |-  ( ph  ->  Y  e.  A )
27 onelon 4433 . . . . . . . . 9  |-  ( ( A  e.  On  /\  Y  e.  A )  ->  Y  e.  On )
282, 26, 27syl2anc 642 . . . . . . . 8  |-  ( ph  ->  Y  e.  On )
29 omcl 6551 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Y  e.  On )  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
3023, 28, 29syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  e.  On )
3120simp3d 969 . . . . . . . 8  |-  ( ph  ->  Z  e.  ( A  ^o  X ) )
32 onelon 4433 . . . . . . . 8  |-  ( ( ( A  ^o  X
)  e.  On  /\  Z  e.  ( A  ^o  X ) )  ->  Z  e.  On )
3323, 31, 32syl2anc 642 . . . . . . 7  |-  ( ph  ->  Z  e.  On )
34 oaword1 6566 . . . . . . 7  |-  ( ( ( ( A  ^o  X )  .o  Y
)  e.  On  /\  Z  e.  On )  ->  ( ( A  ^o  X )  .o  Y
)  C_  ( (
( A  ^o  X
)  .o  Y )  +o  Z ) )
3530, 33, 34syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( A  ^o  X )  .o  Y
)  C_  ( (
( A  ^o  X
)  .o  Y )  +o  Z ) )
36 dif1o 6515 . . . . . . . . . . 11  |-  ( Y  e.  ( A  \  1o )  <->  ( Y  e.  A  /\  Y  =/=  (/) ) )
3736simprbi 450 . . . . . . . . . 10  |-  ( Y  e.  ( A  \  1o )  ->  Y  =/=  (/) )
3824, 37syl 15 . . . . . . . . 9  |-  ( ph  ->  Y  =/=  (/) )
39 on0eln0 4463 . . . . . . . . . 10  |-  ( Y  e.  On  ->  ( (/) 
e.  Y  <->  Y  =/=  (/) ) )
4028, 39syl 15 . . . . . . . . 9  |-  ( ph  ->  ( (/)  e.  Y  <->  Y  =/=  (/) ) )
4138, 40mpbird 223 . . . . . . . 8  |-  ( ph  -> 
(/)  e.  Y )
42 omword1 6587 . . . . . . . 8  |-  ( ( ( ( A  ^o  X )  e.  On  /\  Y  e.  On )  /\  (/)  e.  Y )  ->  ( A  ^o  X )  C_  (
( A  ^o  X
)  .o  Y ) )
4323, 28, 41, 42syl21anc 1181 . . . . . . 7  |-  ( ph  ->  ( A  ^o  X
)  C_  ( ( A  ^o  X )  .o  Y ) )
4443, 31sseldd 3194 . . . . . 6  |-  ( ph  ->  Z  e.  ( ( A  ^o  X )  .o  Y ) )
4535, 44sseldd 3194 . . . . 5  |-  ( ph  ->  Z  e.  ( ( ( A  ^o  X
)  .o  Y )  +o  Z ) )
4619simprd 449 . . . . 5  |-  ( ph  ->  ( ( ( A  ^o  X )  .o  Y )  +o  Z
)  =  C )
4745, 46eleqtrd 2372 . . . 4  |-  ( ph  ->  Z  e.  C )
481, 47sseldd 3194 . . 3  |-  ( ph  ->  Z  e.  ran  ( A CNF  B ) )
493, 2, 4cantnff 7391 . . . 4  |-  ( ph  ->  ( A CNF  B ) : S --> ( A  ^o  B ) )
50 ffn 5405 . . . 4  |-  ( ( A CNF  B ) : S --> ( A  ^o  B )  ->  ( A CNF  B )  Fn  S
)
51 fvelrnb 5586 . . . 4  |-  ( ( A CNF  B )  Fn  S  ->  ( Z  e.  ran  ( A CNF  B
)  <->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z ) )
5249, 50, 513syl 18 . . 3  |-  ( ph  ->  ( Z  e.  ran  ( A CNF  B )  <->  E. g  e.  S  ( ( A CNF  B ) `
 g )  =  Z ) )
5348, 52mpbid 201 . 2  |-  ( ph  ->  E. g  e.  S  ( ( A CNF  B
) `  g )  =  Z )
542adantr 451 . . . . 5  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  A  e.  On )
554adantr 451 . . . . 5  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  B  e.  On )
566adantr 451 . . . . 5  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ( A  ^o  B ) )
571adantr 451 . . . . 5  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  C_  ran  ( A CNF 
B ) )
587adantr 451 . . . . 5  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  (/) 
e.  C )
59 simprl 732 . . . . 5  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
g  e.  S )
60 simprr 733 . . . . 5  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  -> 
( ( A CNF  B
) `  g )  =  Z )
61 eqid 2296 . . . . 5  |-  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( g `  t ) ) )  =  ( t  e.  B  |->  if ( t  =  X ,  Y ,  ( g `  t ) ) )
623, 54, 55, 5, 56, 57, 58, 13, 14, 15, 16, 59, 60, 61cantnflem3 7409 . . . 4  |-  ( (
ph  /\  ( g  e.  S  /\  (
( A CNF  B ) `
 g )  =  Z ) )  ->  C  e.  ran  ( A CNF 
B ) )
6362expr 598 . . 3  |-  ( (
ph  /\  g  e.  S )  ->  (
( ( A CNF  B
) `  g )  =  Z  ->  C  e. 
ran  ( A CNF  B
) ) )
6463rexlimdva 2680 . 2  |-  ( ph  ->  ( E. g  e.  S  ( ( A CNF 
B ) `  g
)  =  Z  ->  C  e.  ran  ( A CNF 
B ) ) )
6553, 64mpd 14 1  |-  ( ph  ->  C  e.  ran  ( A CNF  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {crab 2560    \ cdif 3162    C_ wss 3165   (/)c0 3468   ifcif 3578   <.cop 3656   U.cuni 3843   |^|cint 3878   {copab 4092    e. cmpt 4093   Oncon0 4408   dom cdm 4705   ran crn 4706   iotacio 5233    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   1oc1o 6488   2oc2o 6489    +o coa 6492    .o comu 6493    ^o coe 6494   CNF ccnf 7378
This theorem is referenced by:  cantnf  7411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-cnf 7379
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