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Theorem finxpreclem5 31857
Description: Lemma for  ^^ ^^ recursion theorems. (Contributed by ML, 24-Oct-2020.)
Hypothesis
Ref Expression
finxpreclem5.1  |-  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
finxpreclem5  |-  ( ( n  e.  om  /\  1o  e.  n )  -> 
( -.  x  e.  ( _V  X.  U
)  ->  ( F `  <. n ,  x >. )  =  <. n ,  x >. ) )
Distinct variable group:    x, n
Allowed substitution hints:    U( x, n)    F( x, n)

Proof of Theorem finxpreclem5
StepHypRef Expression
1 df-ov 6311 . . 3  |-  ( n F x )  =  ( F `  <. n ,  x >. )
2 vex 3034 . . . . . 6  |-  x  e. 
_V
3 0ex 4528 . . . . . . 7  |-  (/)  e.  _V
4 opex 4664 . . . . . . . 8  |-  <. U. n ,  ( 1st `  x
) >.  e.  _V
5 opex 4664 . . . . . . . 8  |-  <. n ,  x >.  e.  _V
64, 5ifex 3940 . . . . . . 7  |-  if ( x  e.  ( _V 
X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. )  e.  _V
73, 6ifex 3940 . . . . . 6  |-  if ( ( n  =  1o 
/\  x  e.  U
) ,  (/) ,  if ( x  e.  ( _V  X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) )  e. 
_V
8 finxpreclem5.1 . . . . . . 7  |-  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 >. ) ) )
98ovmpt4g 6438 . . . . . 6  |-  ( ( n  e.  om  /\  x  e.  _V  /\  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) )  e. 
_V )  ->  (
n F x )  =  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V 
X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) ) )
102, 7, 9mp3an23 1382 . . . . 5  |-  ( n  e.  om  ->  (
n F x )  =  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V 
X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) ) )
1110ad2antrr 740 . . . 4  |-  ( ( ( n  e.  om  /\  1o  e.  n )  /\  -.  x  e.  ( _V  X.  U
) )  ->  (
n F x )  =  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V 
X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) ) )
12 1on 7207 . . . . . . . . . . 11  |-  1o  e.  On
1312onirri 5536 . . . . . . . . . 10  |-  -.  1o  e.  1o
14 eleq2 2538 . . . . . . . . . 10  |-  ( n  =  1o  ->  ( 1o  e.  n  <->  1o  e.  1o ) )
1513, 14mtbiri 310 . . . . . . . . 9  |-  ( n  =  1o  ->  -.  1o  e.  n )
1615con2i 124 . . . . . . . 8  |-  ( 1o  e.  n  ->  -.  n  =  1o )
1716intnanrd 931 . . . . . . 7  |-  ( 1o  e.  n  ->  -.  ( n  =  1o  /\  x  e.  U ) )
1817iffalsed 3883 . . . . . 6  |-  ( 1o  e.  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 >. ) )
1918adantl 473 . . . . 5  |-  ( ( n  e.  om  /\  1o  e.  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 >. ) )
20 iffalse 3881 . . . . 5  |-  ( -.  x  e.  ( _V 
X.  U )  ->  if ( x  e.  ( _V  X.  U ) ,  <. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. )  =  <. n ,  x >. )
2119, 20sylan9eq 2525 . . . 4  |-  ( ( ( n  e.  om  /\  1o  e.  n )  /\  -.  x  e.  ( _V  X.  U
) )  ->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) )  = 
<. n ,  x >. )
2211, 21eqtrd 2505 . . 3  |-  ( ( ( n  e.  om  /\  1o  e.  n )  /\  -.  x  e.  ( _V  X.  U
) )  ->  (
n F x )  =  <. n ,  x >. )
231, 22syl5eqr 2519 . 2  |-  ( ( ( n  e.  om  /\  1o  e.  n )  /\  -.  x  e.  ( _V  X.  U
) )  ->  ( F `  <. n ,  x >. )  =  <. n ,  x >. )
2423ex 441 1  |-  ( ( n  e.  om  /\  1o  e.  n )  -> 
( -.  x  e.  ( _V  X.  U
)  ->  ( F `  <. n ,  x >. )  =  <. n ,  x >. ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   (/)c0 3722   ifcif 3872   <.cop 3965   U.cuni 4190    X. cxp 4837   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   omcom 6711   1stc1st 6810   1oc1o 7193
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
This theorem is referenced by:  finxpreclem6  31858
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