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Theorem finxpreclem3 31855
Description: Lemma for  ^^ ^^ recursion theorems. (Contributed by ML, 20-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem3.1  |-  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
finxpreclem3  |-  ( ( ( N  e.  om  /\  2o  C_  N )  /\  X  e.  ( _V  X.  U ) )  ->  <. U. N ,  ( 1st `  X )
>.  =  ( F `  <. N ,  X >. ) )
Distinct variable groups:    n, N, x    U, n, x    n, X, x
Allowed substitution hints:    F( x, n)

Proof of Theorem finxpreclem3
StepHypRef Expression
1 df-ov 6311 . 2  |-  ( N F X )  =  ( F `  <. N ,  X >. )
2 finxpreclem3.1 . . . 4  |-  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 >. ) ) )
32a1i 11 . . 3  |-  ( ( ( N  e.  om  /\  2o  C_  N )  /\  X  e.  ( _V  X.  U ) )  ->  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 >. ) ) ) )
4 eqeq1 2475 . . . . . . 7  |-  ( n  =  N  ->  (
n  =  1o  <->  N  =  1o ) )
5 eleq1 2537 . . . . . . 7  |-  ( x  =  X  ->  (
x  e.  U  <->  X  e.  U ) )
64, 5bi2anan9 890 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  ->  ( ( n  =  1o  /\  x  e.  U )  <->  ( N  =  1o  /\  X  e.  U ) ) )
7 eleq1 2537 . . . . . . . 8  |-  ( x  =  X  ->  (
x  e.  ( _V 
X.  U )  <->  X  e.  ( _V  X.  U
) ) )
87adantl 473 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  ->  ( x  e.  ( _V  X.  U )  <-> 
X  e.  ( _V 
X.  U ) ) )
9 unieq 4198 . . . . . . . . 9  |-  ( n  =  N  ->  U. n  =  U. N )
109adantr 472 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  U. n  =  U. N )
11 fveq2 5879 . . . . . . . . 9  |-  ( x  =  X  ->  ( 1st `  x )  =  ( 1st `  X
) )
1211adantl 473 . . . . . . . 8  |-  ( ( n  =  N  /\  x  =  X )  ->  ( 1st `  x
)  =  ( 1st `  X ) )
1310, 12opeq12d 4166 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  -> 
<. U. n ,  ( 1st `  x )
>.  =  <. U. N ,  ( 1st `  X
) >. )
14 opeq12 4160 . . . . . . 7  |-  ( ( n  =  N  /\  x  =  X )  -> 
<. n ,  x >.  = 
<. N ,  X >. )
158, 13, 14ifbieq12d 3899 . . . . . 6  |-  ( ( n  =  N  /\  x  =  X )  ->  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. )  =  if ( X  e.  ( _V  X.  U ) ,  <. U. N ,  ( 1st `  X )
>. ,  <. N ,  X >. ) )
166, 15ifbieq2d 3897 . . . . 5  |-  ( ( n  =  N  /\  x  =  X )  ->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) )  =  if ( ( N  =  1o  /\  X  e.  U ) ,  (/) ,  if ( X  e.  ( _V  X.  U
) ,  <. U. N ,  ( 1st `  X
) >. ,  <. N ,  X >. ) ) )
17 sssucid 5507 . . . . . . . . . . . . 13  |-  1o  C_  suc  1o
18 df-2o 7201 . . . . . . . . . . . . 13  |-  2o  =  suc  1o
1917, 18sseqtr4i 3451 . . . . . . . . . . . 12  |-  1o  C_  2o
20 1on 7207 . . . . . . . . . . . . . 14  |-  1o  e.  On
2118, 20sucneqoni 31839 . . . . . . . . . . . . 13  |-  2o  =/=  1o
2221necomi 2697 . . . . . . . . . . . 12  |-  1o  =/=  2o
23 df-pss 3406 . . . . . . . . . . . 12  |-  ( 1o  C.  2o  <->  ( 1o  C_  2o  /\  1o  =/=  2o ) )
2419, 22, 23mpbir2an 934 . . . . . . . . . . 11  |-  1o  C.  2o
25 ssnpss 3522 . . . . . . . . . . 11  |-  ( 2o  C_  1o  ->  -.  1o  C.  2o )
2624, 25mt2 184 . . . . . . . . . 10  |-  -.  2o  C_  1o
27 sseq2 3440 . . . . . . . . . 10  |-  ( N  =  1o  ->  ( 2o  C_  N  <->  2o  C_  1o ) )
2826, 27mtbiri 310 . . . . . . . . 9  |-  ( N  =  1o  ->  -.  2o  C_  N )
2928con2i 124 . . . . . . . 8  |-  ( 2o  C_  N  ->  -.  N  =  1o )
3029intnanrd 931 . . . . . . 7  |-  ( 2o  C_  N  ->  -.  ( N  =  1o  /\  X  e.  U ) )
3130iffalsed 3883 . . . . . 6  |-  ( 2o  C_  N  ->  if ( ( N  =  1o 
/\  X  e.  U
) ,  (/) ,  if ( X  e.  ( _V  X.  U ) , 
<. U. N ,  ( 1st `  X )
>. ,  <. N ,  X >. ) )  =  if ( X  e.  ( _V  X.  U
) ,  <. U. N ,  ( 1st `  X
) >. ,  <. N ,  X >. ) )
32 iftrue 3878 . . . . . 6  |-  ( X  e.  ( _V  X.  U )  ->  if ( X  e.  ( _V  X.  U ) , 
<. U. N ,  ( 1st `  X )
>. ,  <. N ,  X >. )  =  <. U. N ,  ( 1st `  X ) >. )
3331, 32sylan9eq 2525 . . . . 5  |-  ( ( 2o  C_  N  /\  X  e.  ( _V  X.  U ) )  ->  if ( ( N  =  1o  /\  X  e.  U ) ,  (/) ,  if ( X  e.  ( _V  X.  U
) ,  <. U. N ,  ( 1st `  X
) >. ,  <. N ,  X >. ) )  = 
<. U. N ,  ( 1st `  X )
>. )
3416, 33sylan9eqr 2527 . . . 4  |-  ( ( ( 2o  C_  N  /\  X  e.  ( _V  X.  U ) )  /\  ( n  =  N  /\  x  =  X ) )  ->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) )  = 
<. U. N ,  ( 1st `  X )
>. )
3534adantlll 732 . . 3  |-  ( ( ( ( N  e. 
om  /\  2o  C_  N
)  /\  X  e.  ( _V  X.  U
) )  /\  (
n  =  N  /\  x  =  X )
)  ->  if (
( n  =  1o 
/\  x  e.  U
) ,  (/) ,  if ( x  e.  ( _V  X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) )  = 
<. U. N ,  ( 1st `  X )
>. )
36 simpll 768 . . 3  |-  ( ( ( N  e.  om  /\  2o  C_  N )  /\  X  e.  ( _V  X.  U ) )  ->  N  e.  om )
37 elex 3040 . . . 4  |-  ( X  e.  ( _V  X.  U )  ->  X  e.  _V )
3837adantl 473 . . 3  |-  ( ( ( N  e.  om  /\  2o  C_  N )  /\  X  e.  ( _V  X.  U ) )  ->  X  e.  _V )
39 opex 4664 . . . 4  |-  <. U. N ,  ( 1st `  X
) >.  e.  _V
4039a1i 11 . . 3  |-  ( ( ( N  e.  om  /\  2o  C_  N )  /\  X  e.  ( _V  X.  U ) )  ->  <. U. N ,  ( 1st `  X )
>.  e.  _V )
413, 35, 36, 38, 40ovmpt2d 6443 . 2  |-  ( ( ( N  e.  om  /\  2o  C_  N )  /\  X  e.  ( _V  X.  U ) )  ->  ( N F X )  =  <. U. N ,  ( 1st `  X ) >. )
421, 41syl5reqr 2520 1  |-  ( ( ( N  e.  om  /\  2o  C_  N )  /\  X  e.  ( _V  X.  U ) )  ->  <. U. N ,  ( 1st `  X )
>.  =  ( F `  <. N ,  X >. ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031    C_ wss 3390    C. wpss 3391   (/)c0 3722   ifcif 3872   <.cop 3965   U.cuni 4190    X. cxp 4837   suc csuc 5432   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   omcom 6711   1stc1st 6810   1oc1o 7193   2oc2o 7194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1o 7200  df-2o 7201
This theorem is referenced by:  finxpreclem4  31856
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