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Theorem finxpsuclem 31754
Description: Lemma for finxpsuc 31755. (Contributed by ML, 24-Oct-2020.)
Hypothesis
Ref Expression
finxpsuclem.1  |-  F  =  ( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) ) )
Assertion
Ref Expression
finxpsuclem  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( U ^^ ^^ suc  N )  =  ( ( U ^^ ^^ N )  X.  U
) )
Distinct variable groups:    n, N, x    U, n, x
Allowed substitution hints:    F( x, n)

Proof of Theorem finxpsuclem
Dummy variables  z 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 peano2 6728 . . . . . . . . . 10  |-  ( N  e.  om  ->  suc  N  e.  om )
21adantr 466 . . . . . . . . 9  |-  ( ( N  e.  om  /\  1o  C_  N )  ->  suc  N  e.  om )
3 1on 7201 . . . . . . . . . . . . 13  |-  1o  e.  On
43onordi 5546 . . . . . . . . . . . 12  |-  Ord  1o
5 nnord 6715 . . . . . . . . . . . 12  |-  ( N  e.  om  ->  Ord  N )
6 ordsseleq 5471 . . . . . . . . . . . 12  |-  ( ( Ord  1o  /\  Ord  N )  ->  ( 1o  C_  N  <->  ( 1o  e.  N  \/  1o  =  N ) ) )
74, 5, 6sylancr 667 . . . . . . . . . . 11  |-  ( N  e.  om  ->  ( 1o  C_  N  <->  ( 1o  e.  N  \/  1o  =  N ) ) )
87biimpa 486 . . . . . . . . . 10  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( 1o  e.  N  \/  1o  =  N ) )
9 elelsuc 5514 . . . . . . . . . . . . 13  |-  ( 1o  e.  N  ->  1o  e.  suc  N )
109a1i 11 . . . . . . . . . . . 12  |-  ( N  e.  om  ->  ( 1o  e.  N  ->  1o  e.  suc  N ) )
11 sucidg 5520 . . . . . . . . . . . . 13  |-  ( N  e.  om  ->  N  e.  suc  N )
12 eleq1 2495 . . . . . . . . . . . . 13  |-  ( 1o  =  N  ->  ( 1o  e.  suc  N  <->  N  e.  suc  N ) )
1311, 12syl5ibrcom 225 . . . . . . . . . . . 12  |-  ( N  e.  om  ->  ( 1o  =  N  ->  1o  e.  suc  N ) )
1410, 13jaod 381 . . . . . . . . . . 11  |-  ( N  e.  om  ->  (
( 1o  e.  N  \/  1o  =  N )  ->  1o  e.  suc  N ) )
1514adantr 466 . . . . . . . . . 10  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( ( 1o  e.  N  \/  1o  =  N )  ->  1o  e.  suc  N ) )
168, 15mpd 15 . . . . . . . . 9  |-  ( ( N  e.  om  /\  1o  C_  N )  ->  1o  e.  suc  N )
17 finxpsuclem.1 . . . . . . . . . 10  |-  F  =  ( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) ) )
1817finxpreclem6 31753 . . . . . . . . 9  |-  ( ( suc  N  e.  om  /\  1o  e.  suc  N
)  ->  ( U ^^ ^^ suc  N ) 
C_  ( _V  X.  U ) )
192, 16, 18syl2anc 665 . . . . . . . 8  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( U ^^ ^^ suc  N )  C_  ( _V  X.  U ) )
2019sselda 3464 . . . . . . 7  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( U ^^ ^^ suc  N
) )  ->  y  e.  ( _V  X.  U
) )
211ad2antrr 730 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  suc  N  e.  om )
22 df-2o 7195 . . . . . . . . . . . . . . 15  |-  2o  =  suc  1o
23 ordsucsssuc 6665 . . . . . . . . . . . . . . . . 17  |-  ( ( Ord  1o  /\  Ord  N )  ->  ( 1o  C_  N  <->  suc  1o  C_  suc  N ) )
244, 5, 23sylancr 667 . . . . . . . . . . . . . . . 16  |-  ( N  e.  om  ->  ( 1o  C_  N  <->  suc  1o  C_  suc  N ) )
2524biimpa 486 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  om  /\  1o  C_  N )  ->  suc  1o  C_  suc  N )
2622, 25syl5eqss 3508 . . . . . . . . . . . . . 14  |-  ( ( N  e.  om  /\  1o  C_  N )  ->  2o  C_  suc  N )
2726adantr 466 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  2o  C_  suc  N )
28 simpr 462 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  y  e.  ( _V  X.  U ) )
2917finxpreclem4 31751 . . . . . . . . . . . . 13  |-  ( ( ( suc  N  e. 
om  /\  2o  C_  suc  N )  /\  y  e.  ( _V  X.  U
) )  ->  ( rec ( F ,  <. suc 
N ,  y >.
) `  suc  N )  =  ( rec ( F ,  <. U. suc  N ,  ( 1st `  y
) >. ) `  U. suc  N ) )
3021, 27, 28, 29syl21anc 1263 . . . . . . . . . . . 12  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( rec ( F ,  <. suc  N ,  y >. ) `  suc  N )  =  ( rec ( F ,  <. U. suc  N , 
( 1st `  y
) >. ) `  U. suc  N ) )
31 ordunisuc 6674 . . . . . . . . . . . . . . . 16  |-  ( Ord 
N  ->  U. suc  N  =  N )
325, 31syl 17 . . . . . . . . . . . . . . 15  |-  ( N  e.  om  ->  U. suc  N  =  N )
33 opeq1 4187 . . . . . . . . . . . . . . . 16  |-  ( U. suc  N  =  N  ->  <. U. suc  N , 
( 1st `  y
) >.  =  <. N , 
( 1st `  y
) >. )
34 rdgeq2 7142 . . . . . . . . . . . . . . . 16  |-  ( <. U. suc  N ,  ( 1st `  y )
>.  =  <. N , 
( 1st `  y
) >.  ->  rec ( F ,  <. U. suc  N ,  ( 1st `  y
) >. )  =  rec ( F ,  <. N , 
( 1st `  y
) >. ) )
3533, 34syl 17 . . . . . . . . . . . . . . 15  |-  ( U. suc  N  =  N  ->  rec ( F ,  <. U.
suc  N ,  ( 1st `  y )
>. )  =  rec ( F ,  <. N , 
( 1st `  y
) >. ) )
3632, 35syl 17 . . . . . . . . . . . . . 14  |-  ( N  e.  om  ->  rec ( F ,  <. U. suc  N ,  ( 1st `  y
) >. )  =  rec ( F ,  <. N , 
( 1st `  y
) >. ) )
3736, 32fveq12d 5888 . . . . . . . . . . . . 13  |-  ( N  e.  om  ->  ( rec ( F ,  <. U.
suc  N ,  ( 1st `  y )
>. ) `  U. suc  N )  =  ( rec ( F ,  <. N ,  ( 1st `  y
) >. ) `  N
) )
3837ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( rec ( F ,  <. U. suc  N ,  ( 1st `  y
) >. ) `  U. suc  N )  =  ( rec ( F ,  <. N ,  ( 1st `  y ) >. ) `  N ) )
3930, 38eqtrd 2463 . . . . . . . . . . 11  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( rec ( F ,  <. suc  N ,  y >. ) `  suc  N )  =  ( rec ( F ,  <. N ,  ( 1st `  y )
>. ) `  N ) )
4039eqeq2d 2436 . . . . . . . . . 10  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( (/)  =  ( rec ( F ,  <. suc  N ,  y
>. ) `  suc  N
)  <->  (/)  =  ( rec ( F ,  <. N ,  ( 1st `  y
) >. ) `  N
) ) )
411biantrurd 510 . . . . . . . . . . . 12  |-  ( N  e.  om  ->  ( (/)  =  ( rec ( F ,  <. suc  N ,  y >. ) `  suc  N )  <->  ( suc  N  e.  om  /\  (/)  =  ( rec ( F ,  <. suc  N ,  y
>. ) `  suc  N
) ) ) )
4217dffinxpf 31742 . . . . . . . . . . . . 13  |-  ( U ^^ ^^ suc  N
)  =  { y  |  ( suc  N  e.  om  /\  (/)  =  ( rec ( F ,  <. suc  N ,  y
>. ) `  suc  N
) ) }
4342abeq2i 2544 . . . . . . . . . . . 12  |-  ( y  e.  ( U ^^ ^^ suc  N )  <->  ( suc  N  e.  om  /\  (/)  =  ( rec ( F ,  <. suc  N ,  y
>. ) `  suc  N
) ) )
4441, 43syl6rbbr 267 . . . . . . . . . . 11  |-  ( N  e.  om  ->  (
y  e.  ( U ^^ ^^ suc  N
)  <->  (/)  =  ( rec ( F ,  <. suc 
N ,  y >.
) `  suc  N ) ) )
4544ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( y  e.  ( U ^^ ^^ suc  N )  <->  (/)  =  ( rec ( F ,  <. suc  N ,  y
>. ) `  suc  N
) ) )
46 fvex 5892 . . . . . . . . . . . . 13  |-  ( 1st `  y )  e.  _V
47 opeq2 4188 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( 1st `  y
)  ->  <. N , 
z >.  =  <. N , 
( 1st `  y
) >. )
48 rdgeq2 7142 . . . . . . . . . . . . . . . . 17  |-  ( <. N ,  z >.  = 
<. N ,  ( 1st `  y ) >.  ->  rec ( F ,  <. N , 
z >. )  =  rec ( F ,  <. N , 
( 1st `  y
) >. ) )
4947, 48syl 17 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( 1st `  y
)  ->  rec ( F ,  <. N , 
z >. )  =  rec ( F ,  <. N , 
( 1st `  y
) >. ) )
5049fveq1d 5884 . . . . . . . . . . . . . . 15  |-  ( z  =  ( 1st `  y
)  ->  ( rec ( F ,  <. N , 
z >. ) `  N
)  =  ( rec ( F ,  <. N ,  ( 1st `  y
) >. ) `  N
) )
5150eqeq2d 2436 . . . . . . . . . . . . . 14  |-  ( z  =  ( 1st `  y
)  ->  ( (/)  =  ( rec ( F ,  <. N ,  z >.
) `  N )  <->  (/)  =  ( rec ( F ,  <. N , 
( 1st `  y
) >. ) `  N
) ) )
5251anbi2d 708 . . . . . . . . . . . . 13  |-  ( z  =  ( 1st `  y
)  ->  ( ( N  e.  om  /\  (/)  =  ( rec ( F ,  <. N ,  z >.
) `  N )
)  <->  ( N  e. 
om  /\  (/)  =  ( rec ( F ,  <. N ,  ( 1st `  y ) >. ) `  N ) ) ) )
5317dffinxpf 31742 . . . . . . . . . . . . 13  |-  ( U ^^ ^^ N )  =  { z  |  ( N  e.  om  /\  (/)  =  ( rec ( F ,  <. N , 
z >. ) `  N
) ) }
5446, 52, 53elab2 3220 . . . . . . . . . . . 12  |-  ( ( 1st `  y )  e.  ( U ^^ ^^ N )  <->  ( N  e.  om  /\  (/)  =  ( rec ( F ,  <. N ,  ( 1st `  y ) >. ) `  N ) ) )
5554baib 911 . . . . . . . . . . 11  |-  ( N  e.  om  ->  (
( 1st `  y
)  e.  ( U ^^ ^^ N )  <->  (/)  =  ( rec ( F ,  <. N , 
( 1st `  y
) >. ) `  N
) ) )
5655ad2antrr 730 . . . . . . . . . 10  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( ( 1st `  y )  e.  ( U ^^ ^^ N
)  <->  (/)  =  ( rec ( F ,  <. N ,  ( 1st `  y
) >. ) `  N
) ) )
5740, 45, 563bitr4d 288 . . . . . . . . 9  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( y  e.  ( U ^^ ^^ suc  N )  <->  ( 1st `  y )  e.  ( U ^^ ^^ N
) ) )
5857biimpd 210 . . . . . . . 8  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( _V  X.  U ) )  ->  ( y  e.  ( U ^^ ^^ suc  N )  ->  ( 1st `  y )  e.  ( U ^^ ^^ N ) ) )
5958impancom 441 . . . . . . 7  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( U ^^ ^^ suc  N
) )  ->  (
y  e.  ( _V 
X.  U )  -> 
( 1st `  y
)  e.  ( U ^^ ^^ N ) ) )
6020, 59mpd 15 . . . . . 6  |-  ( ( ( N  e.  om  /\  1o  C_  N )  /\  y  e.  ( U ^^ ^^ suc  N
) )  ->  ( 1st `  y )  e.  ( U ^^ ^^ N ) )
6160ex 435 . . . . 5  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( y  e.  ( U ^^ ^^ suc  N )  ->  ( 1st `  y )  e.  ( U ^^ ^^ N
) ) )
6220ex 435 . . . . 5  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( y  e.  ( U ^^ ^^ suc  N )  ->  y  e.  ( _V  X.  U
) ) )
6361, 62jcad 535 . . . 4  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( y  e.  ( U ^^ ^^ suc  N )  ->  ( ( 1st `  y )  e.  ( U ^^ ^^ N )  /\  y  e.  ( _V  X.  U
) ) ) )
6457exbiri 626 . . . . . 6  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( y  e.  ( _V  X.  U )  ->  ( ( 1st `  y )  e.  ( U ^^ ^^ N
)  ->  y  e.  ( U ^^ ^^ suc  N ) ) ) )
6564impd 432 . . . . 5  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( ( y  e.  ( _V  X.  U
)  /\  ( 1st `  y )  e.  ( U ^^ ^^ N
) )  ->  y  e.  ( U ^^ ^^ suc  N ) ) )
6665ancomsd 455 . . . 4  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( ( ( 1st `  y )  e.  ( U ^^ ^^ N
)  /\  y  e.  ( _V  X.  U
) )  ->  y  e.  ( U ^^ ^^ suc  N ) ) )
6763, 66impbid 193 . . 3  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( y  e.  ( U ^^ ^^ suc  N )  <->  ( ( 1st `  y )  e.  ( U ^^ ^^ N
)  /\  y  e.  ( _V  X.  U
) ) ) )
68 elxp8 31739 . . 3  |-  ( y  e.  ( ( U ^^ ^^ N )  X.  U )  <->  ( ( 1st `  y )  e.  ( U ^^ ^^ N )  /\  y  e.  ( _V  X.  U
) ) )
6967, 68syl6bbr 266 . 2  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( y  e.  ( U ^^ ^^ suc  N )  <->  y  e.  ( ( U ^^ ^^ N )  X.  U
) ) )
7069eqrdv 2419 1  |-  ( ( N  e.  om  /\  1o  C_  N )  -> 
( U ^^ ^^ suc  N )  =  ( ( U ^^ ^^ N )  X.  U
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   _Vcvv 3080    C_ wss 3436   (/)c0 3761   ifcif 3911   <.cop 4004   U.cuni 4219    X. cxp 4851   Ord word 5441   suc csuc 5444   ` cfv 5601    |-> cmpt2 6308   omcom 6707   1stc1st 6806   reccrdg 7139   1oc1o 7187   2oc2o 7188   ^^
^^cfinxp 31740
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-reg 8117
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-int 4256  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-1o 7194  df-2o 7195  df-oadd 7198  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-fin 7585  df-finxp 31741
This theorem is referenced by:  finxpsuc  31755
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