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Theorem fseqdom 7842
Description: One half of fseqen 7843. (Contributed by Mario Carneiro, 18-Nov-2014.)
Assertion
Ref Expression
fseqdom  |-  ( A  e.  V  ->  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )
Distinct variable group:    A, n
Allowed substitution hint:    V( n)

Proof of Theorem fseqdom
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 7533 . . 3  |-  om  e.  _V
2 ovex 6047 . . 3  |-  ( A  ^m  n )  e. 
_V
31, 2iunex 5932 . 2  |-  U_ n  e.  om  ( A  ^m  n )  e.  _V
4 xp1st 6317 . . . . . . . 8  |-  ( x  e.  ( om  X.  A )  ->  ( 1st `  x )  e. 
om )
5 peano2 4807 . . . . . . . 8  |-  ( ( 1st `  x )  e.  om  ->  suc  ( 1st `  x )  e.  om )
64, 5syl 16 . . . . . . 7  |-  ( x  e.  ( om  X.  A )  ->  suc  ( 1st `  x )  e.  om )
76adantl 453 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  ->  suc  ( 1st `  x
)  e.  om )
8 xp2nd 6318 . . . . . . . . 9  |-  ( x  e.  ( om  X.  A )  ->  ( 2nd `  x )  e.  A )
98adantl 453 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( 2nd `  x
)  e.  A )
10 fconst6g 5574 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  A  ->  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } ) : suc  ( 1st `  x ) --> A )
119, 10syl 16 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A )
12 elmapg 6969 . . . . . . . 8  |-  ( ( A  e.  V  /\  suc  ( 1st `  x
)  e.  om )  ->  ( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) )  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A ) )
136, 12sylan2 461 . . . . . . 7  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) )  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } ) : suc  ( 1st `  x
) --> A ) )
1411, 13mpbird 224 . . . . . 6  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) )
15 oveq2 6030 . . . . . . . 8  |-  ( n  =  suc  ( 1st `  x )  ->  ( A  ^m  n )  =  ( A  ^m  suc  ( 1st `  x ) ) )
1615eleq2d 2456 . . . . . . 7  |-  ( n  =  suc  ( 1st `  x )  ->  (
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e.  ( A  ^m  n
)  <->  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) ) )
1716rspcev 2997 . . . . . 6  |-  ( ( suc  ( 1st `  x
)  e.  om  /\  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  ( A  ^m  suc  ( 1st `  x ) ) )  ->  E. n  e.  om  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  n
) )
187, 14, 17syl2anc 643 . . . . 5  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  ->  E. n  e.  om  ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  ( A  ^m  n ) )
19 eliun 4041 . . . . 5  |-  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  e.  U_ n  e.  om  ( A  ^m  n )  <->  E. n  e.  om  ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  e.  ( A  ^m  n
) )
2018, 19sylibr 204 . . . 4  |-  ( ( A  e.  V  /\  x  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e. 
U_ n  e.  om  ( A  ^m  n
) )
2120ex 424 . . 3  |-  ( A  e.  V  ->  (
x  e.  ( om 
X.  A )  -> 
( suc  ( 1st `  x )  X.  {
( 2nd `  x
) } )  e. 
U_ n  e.  om  ( A  ^m  n
) ) )
22 nsuceq0 4604 . . . . . . 7  |-  suc  ( 1st `  x )  =/=  (/)
23 fvex 5684 . . . . . . . 8  |-  ( 2nd `  x )  e.  _V
2423snnz 3867 . . . . . . 7  |-  { ( 2nd `  x ) }  =/=  (/)
25 xp11 5246 . . . . . . 7  |-  ( ( suc  ( 1st `  x
)  =/=  (/)  /\  {
( 2nd `  x
) }  =/=  (/) )  -> 
( ( suc  ( 1st `  x )  X. 
{ ( 2nd `  x
) } )  =  ( suc  ( 1st `  y )  X.  {
( 2nd `  y
) } )  <->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y )  /\  {
( 2nd `  x
) }  =  {
( 2nd `  y
) } ) ) )
2622, 24, 25mp2an 654 . . . . . 6  |-  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y )  /\  {
( 2nd `  x
) }  =  {
( 2nd `  y
) } ) )
27 xp1st 6317 . . . . . . . . 9  |-  ( y  e.  ( om  X.  A )  ->  ( 1st `  y )  e. 
om )
28 peano4 4809 . . . . . . . . 9  |-  ( ( ( 1st `  x
)  e.  om  /\  ( 1st `  y )  e.  om )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
294, 27, 28syl2an 464 . . . . . . . 8  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( suc  ( 1st `  x )  =  suc  ( 1st `  y )  <-> 
( 1st `  x
)  =  ( 1st `  y ) ) )
3029adantl 453 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( suc  ( 1st `  x )  =  suc  ( 1st `  y
)  <->  ( 1st `  x
)  =  ( 1st `  y ) ) )
31 sneqbg 3913 . . . . . . . 8  |-  ( ( 2nd `  x )  e.  _V  ->  ( { ( 2nd `  x
) }  =  {
( 2nd `  y
) }  <->  ( 2nd `  x )  =  ( 2nd `  y ) ) )
3223, 31mp1i 12 . . . . . . 7  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( { ( 2nd `  x ) }  =  { ( 2nd `  y ) }  <->  ( 2nd `  x
)  =  ( 2nd `  y ) ) )
3330, 32anbi12d 692 . . . . . 6  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  =  suc  ( 1st `  y )  /\  { ( 2nd `  x
) }  =  {
( 2nd `  y
) } )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) ) ) )
3426, 33syl5bb 249 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) ) ) )
35 xpopth 6329 . . . . . 6  |-  ( ( x  e.  ( om 
X.  A )  /\  y  e.  ( om  X.  A ) )  -> 
( ( ( 1st `  x )  =  ( 1st `  y )  /\  ( 2nd `  x
)  =  ( 2nd `  y ) )  <->  x  =  y ) )
3635adantl 453 . . . . 5  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( ( 1st `  x )  =  ( 1st `  y
)  /\  ( 2nd `  x )  =  ( 2nd `  y ) )  <->  x  =  y
) )
3734, 36bitrd 245 . . . 4  |-  ( ( A  e.  V  /\  ( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  x  =  y ) )
3837ex 424 . . 3  |-  ( A  e.  V  ->  (
( x  e.  ( om  X.  A )  /\  y  e.  ( om  X.  A ) )  ->  ( ( suc  ( 1st `  x
)  X.  { ( 2nd `  x ) } )  =  ( suc  ( 1st `  y
)  X.  { ( 2nd `  y ) } )  <->  x  =  y ) ) )
3921, 38dom2d 7086 . 2  |-  ( A  e.  V  ->  ( U_ n  e.  om  ( A  ^m  n
)  e.  _V  ->  ( om  X.  A )  ~<_ 
U_ n  e.  om  ( A  ^m  n
) ) )
403, 39mpi 17 1  |-  ( A  e.  V  ->  ( om  X.  A )  ~<_  U_ n  e.  om  ( A  ^m  n ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2552   E.wrex 2652   _Vcvv 2901   (/)c0 3573   {csn 3759   U_ciun 4037   class class class wbr 4155   suc csuc 4526   omcom 4787    X. cxp 4818   -->wf 5392   ` cfv 5396  (class class class)co 6022   1stc1st 6288   2ndc2nd 6289    ^m cmap 6956    ~<_ cdom 7045
This theorem is referenced by:  fseqen  7843
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-map 6958  df-dom 7049
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